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==More==
 
==More==
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*<font size= 5> [[Table_of_indefinite_integrals_axplusb_and_pxplusq|Integrals with ax+b and px+q]]. </font size>
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* [[Table_of_indefinite_integrals_axplusb_and_pxplusq|Integrals with ax+b and px+q]].  
−
*<font size= 5> [[Table_of_indefinite_integrals_xnplusan|Integrals with <math>x^n+a^n</math>]]
+
* [[Table_of_indefinite_integrals_xnplusan|Integrals with <math>x^n+a^n</math>]]
−
*<font size=5>  [[Table_of_indefinite_integrals_xnminusan|Integrals with <math>x^n-a^n</math>]] </font size>
+
* [[Table_of_indefinite_integrals_xnminusan|Integrals with <math>x^n-a^n</math>]] </font size>
−
*<font size=5> [[Table_of_indefinite_integrals_cosine_sine|Integrals with <math>\cos x</math> and/or <math>\sin x</math>]] </font size>
+
* [[Table_of_indefinite_integrals_cosine_sine|Integrals with <math>\cos x</math> and/or <math>\sin x</math>]]
−
 
+
* [[Table_of_indefinite_integrals_cosine_sine|Integrals with <math>\cos x</math> and/or <math>\sin x</math>]]  
 +
*[[Table_of_indefinite_integrals_cotangent|Integrals with cotangent (cot x)]]
 +
*[[Table_of_indefinite_integrals_oneovercosine|Integrals with 1/cos x]]
 +
*[[Table_of_indefinite_integrals_inversecircularfunctions|Integrals with arccos, arcsin, arctan, arc cot]]
 +
*[[Table_of_indefinite_integrals_exponential|Integrals with <math>e^x</math>]]
 +
*[[Table_of_indefinite_integrals_log|Integrals with <math>\ln x</math>]]
 +
*[[Table_of_indefinite_integrals_sh|Integrals with hyperbolic sine (sh x)]]
 +
*[[Table_of_indefinite_integrals_ch|Integrals with hyperbolic cosine (ch x)]]
 +
*[[Table_of_indefinite_integrals_th|Integrals with hyperbolic tangent (th x)]]
 +
*[[Table_of_indefinite_integrals_coth|Integrals with hyperbolic cotangent (both x)]]
 
{|
 
{|
  
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! style="background-color: rgb(238, 238, 238); background-image: none; background-repeat: repeat; background-attachment: scroll; background-position: 0% 0%; -moz-background-size: auto auto; -moz-background-clip: -moz-initial; -moz-background-origin: -moz-initial; -moz-background-inline-policy: -moz-initial; font-size: 110%;" colspan="2" | 22 Integrals Component tan ax
+
 
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|-
+
 
−
|<math> \int \tan a x d x = - \frac {1}{a} \ln {\cos a x } </math>
+
 
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|-
+
 
−
|<math> \int \tan ^2 a x d x = \frac { \tan ax}{a} - x </math>
+
 
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|-
+
 
−
|<math> \int \tan ^3 a x d x = \frac {\tan^2 ax}{2a}+ \frac{1}{a} \ln {\cos a x}</math>
+
 
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|-
+
 
−
|<math> \int \frac {\tan^n ax }{\cos^2 a x}dx = \frac {\tan^{n+1} a x}{(n+1)a} </math>
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|-
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−
|<math> \int \frac {1}{\cos^2 a x \tan ax }dx = \frac {1}{a} \ln {\tan a x} </math>
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|-
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−
|<math> \int \frac {dx}{ \tan ax } = \frac {1}{a} \ln {\sin a x} </math>
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|-
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|<math> \int x \tan ax dx = \frac {1}{a^2} \left \{\frac{(a x)^3}{3} + \frac{(ax)^5}{15}+ \frac {2(ax)^7}{105} + \cdot \cdot \cdot + \frac {2^{2n}(2^{2n-1})Bn(ax)^{2n-1}}{(2n+1)!} + \cdot \cdot \cdot \right \} </math>
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|-
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−
|<math> \int \frac {\tan ax }{ x } dx = ax + \frac{(a x)^3}{9} + \frac{2(ax)^5}{75} + \cdot \cdot \cdot + \frac {2^{2n}(2^{2n-1})Bn(ax)^{2n-1}}{(2n-1)(2n)!} + \cdot \cdot \cdot  </math>
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−
|-
+
−
|<math> \int x \tan^2 ax dx = \frac {x \tan ax}{a} + \frac {1}{a^2} \ln {\cos a x} - \frac {x^2}{2} </math>
+
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|-
+
−
|<math> \int \frac {dx}{p+q \tan ax} = \frac {px}{p^2+q^2} + \frac {q}{a(p^2+q^2)} \ln {\left( q\sin a x + p \cos ax \right)} </math>
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|-
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−
|<math> \int \tan^n ax dx = \frac {\tan^{n+1}ax}{(n+1)a} -\int \tan^{n-2} a x dx </math>
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|-
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! style="background-color: rgb(238, 238, 238); background-image: none; background-repeat: repeat; background-attachment: scroll; background-position: 0% 0%; -moz-background-size: auto auto; -moz-background-clip: -moz-initial; -moz-background-origin: -moz-initial; -moz-background-inline-policy: -moz-initial; font-size: 110%;" colspan="2" | 23 Integrals Component cot ax
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−
|<math> \int \cot a x d x = \frac {1}{a} \ln {\sin a x } </math>
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|-
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−
|<math> \int \cot ^2 a x d x = -\frac { \cot ax}{a} - x </math>
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|-
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−
|<math> \int \cot ^3 a x d x = - \frac {\cot^2 ax}{2a} - \frac{1}{a} \ln {\sin a x}</math>
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+
−
|<math> \int \frac {\cot^n ax }{\sin^2 a x}dx = -\frac {\cot^{n+1} a x}{(n+1)a} </math>
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−
|<math> \int \frac {dx}{\sin^2 a x \cot ax }= - \frac {1}{a} \ln {\cot a x} </math>
+
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|-
+
−
|<math> \int \frac {dx}{ \cot ax } = -\frac {1}{a} \ln {\cos a x} </math>
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|-
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−
|<math> \int x \cot ax dx = \frac {1}{a^2} \left \{ ax - \frac{(a x)^3}{9} - \cdot \cdot \cdot - \frac {2^{2n}Bn(ax)^{2n+1}}{(2n+1)!} + \cdot \cdot \cdot \right \} </math>
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|-
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|<math> \int \frac {\cot ax } {x} dx = -\frac {1}{ax} - \frac{a x}{3} - \frac{(ax)^3}{135} - \cdot \cdot \cdot - \frac {2^{2n}Bn(ax)^{2n-1}}{(2n-1)(2n)!} - \cdot \cdot \cdot  </math>
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|-
+
−
|<math> \int x \cot^2 ax dx = - \frac {x \cot ax}{a} + \frac {1}{a^2} \ln {\sin a x} - \frac {x^2}{2} </math>
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−
|<math> \int \frac {dx}{p+q \cot ax} = \frac {px}{p^2+q^2} - \frac {q}{a(p^2+q^2)} \ln {\left( p\sin a x + q \cos ax \right)} </math>
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|-
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|<math> \int \cot^n ax dx = -\frac {\cot^{n-1}ax}{(n+1)a} -\int \cot^{n-2} a x dx </math>
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|-
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−
! style="background-color: rgb(238, 238, 238); background-image: none; background-repeat: repeat; background-attachment: scroll; background-position: 0% 0%; -moz-background-size: auto auto; -moz-background-clip: -moz-initial; -moz-background-origin: -moz-initial; -moz-background-inline-policy: -moz-initial; font-size: 110%;" colspan="2" | 24 Integrals Component <math> \frac {1}{\cos ax}</math>
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|<math> \int \frac {dx }{\cos a x}dx = \frac {1}{a} \ln { \left( \frac {1} {\cos ax} + \tan ax \right ) } = \frac {1}{a} \ln { \left( \frac {ax}{2} +\frac {\pi}{4} \right) }</math>
+
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|-
+
−
|<math> \int \frac {dx}{ \cos^2 ax }=  \frac {\tan ax} {a} </math>
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|-
+
−
|<math> \int \frac {1}{ \cos^3 ax }dx = \frac {\tan ax}{2a \cos ax}+ \frac {1}{2a} \ln { \left( \frac{1}{\cos ax}+ {\tan a x} \right) } </math>
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|-
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−
|<math> \int \frac {1}{\cos^n ax} \tan ax dx= \frac{a}{na \cos^nax} </math>
+
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|-
+
−
|<math> \int \cos ax dx = \frac {\sin ax}{a} </math>
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−
|<math> \int \frac {xdx} {\cos ax} = \frac {1}{a^2} \left \{ \frac {(ax)^2}{2}+ \frac{(a x)^4}{8}+\frac {5(ax)^6}{144} + \cdot \cdot \cdot + \frac {En(ax)^{2n+2}}{(2n+2)(2n)!} + \cdot \cdot \cdot \right \} </math>
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|-
+
−
|<math> \int \frac {dx}{x \cos ax } = \ln {x} + \frac {(ax)^2} {4} + \frac{5(a x)^4}{96}+\frac{61(ax)^6 }{4320}+ \cdot \cdot \cdot + \frac {En(ax)^{2n}}{2n(2n)!} + \cdot \cdot \cdot  </math>
+
−
|-
+
−
|<math> \int x \cos^2 ax dx = - \frac {x \cot ax}{a} + \frac {1}{a^2} \ln {\sin a x} - \frac {x^2}{2} </math>
+
−
|-
+
−
|<math> \int \frac {x dx}{ \cos^2 ax} = \frac {x}{a} \tan ax + \frac {1}{a^2} \ln { \cos ax } </math>
+
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|-
+
−
|<math> \int \frac {dx}{q+\frac {p}{\cos ax}}= \frac{x}{q}+\frac{p}{q} \int \frac{dx}{p+q\cos ax} </math>
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|-
+
−
|<math> \int \cos^n ax dx = \frac {\tan ax}{a(n-1)\cos^{n-2}ax} + \frac {n-2}{n-1} \int \frac {dx}{\cos^{n-2} a x } </math>
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! style="background-color: rgb(238, 238, 238); background-image: none; background-repeat: repeat; background-attachment: scroll; background-position: 0% 0%; -moz-background-size: auto auto; -moz-background-clip: -moz-initial; -moz-background-origin: -moz-initial; -moz-background-inline-policy: -moz-initial; font-size: 110%;" colspan="2" | 25 Integrals Component <math> \frac {1}{\sin ax}</math>
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−
|<math> \int \frac {dx }{\sin a x}dx = \frac {1}{a} \ln { \left( \frac {1} {\sin ax} - \cot ax \right ) } = \frac {1}{a} \ln { \tan \frac {ax}{2} } </math>
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|-
+
−
|<math> \int \frac {dx}{ \sin^2 ax }=  -\frac {\cot ax} { a} </math>
+
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|-
+
−
|<math> \int \frac {1}{ \sin^3 ax }dx = -\frac {\cot ax}{2a \sin ax}+ \frac {1}{2a} \ln{ \tan \frac{ax}{2} }</math>
+
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|-
+
−
|<math> \int \frac {\cot ax dx}{\sin^n ax}= -\frac{1}{na \sin^nax} </math>
+
−
|-
+
−
|<math> \int \sin ax dx = -\frac {\cos ax}{a} </math>
+
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|-
+
−
|<math> \int \frac {xdx} {\sin ax} = \frac {1}{a^2} \left \{ ax + \frac {(ax)^3}{18}+ \frac{7(a x)^5}{1800} + \cdot \cdot \cdot + \frac {2(2^{2n-1}-1)Bn(ax)^{2n+1}}{(2n+2)!} + \cdot \cdot \cdot \right \} </math>
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|-
+
−
|<math> \int \frac {dx}{x \sin ax } = -\frac{1}{ax} + \frac {ax} {6} + \frac{7(a x)^3}{1080}+ \cdot \cdot \cdot + \frac {2(2^{2n-1}-1)Bn(ax)^{2n+1}}{(2n-1)(2n)!} + \cdot \cdot \cdot  </math>
+
−
|-
+
−
|<math> \int x \sin^2 ax dx = - \frac {x \cot ax}{a} + \frac {1}{a^2} \ln {\sin a x} - \frac {x^2}{2} </math>
+
−
|-
+
−
|<math> \int \frac {x dx}{ \cos^2 ax} = -\frac {x\cot ax}{a} + \frac {1}{a^2} \ln { \sin ax } </math>
+
−
|-
+
−
|<math> \int \frac {dx}{q+\frac {p}{\sin ax}}= \frac{x}{q}-\frac{p}{q} \int \frac{dx}{p+q\sin ax} </math>
+
−
|-
+
−
|<math> \int \sin^n ax dx = -\frac {\cot ax}{a(n-1)\sin^{n-2}ax} + \frac {n-2}{n-1} \int \frac {dx}{\sin^{n-2} a x } </math>
+
−
|-
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−
! style="background-color: rgb(238, 238, 238); background-image: none; background-repeat: repeat; background-attachment: scroll; background-position: 0% 0%; -moz-background-size: auto auto; -moz-background-clip: -moz-initial; -moz-background-origin: -moz-initial; -moz-background-inline-policy: -moz-initial; font-size: 110%;" colspan="2" | 26 Integrals Component the inverse circular functions
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+
−
|<math> \int \arcsin \frac {x }{ a}dx = x \arcsin \frac {x}{a} + \sqrt { a^2-x^2 } </math>
+
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|-
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|<math> \int x \arcsin \frac {x }{ a}dx = \left ( \frac{x^2}{2}-\frac {a^2}{4} \right) \arcsin \frac {x}{a} + \frac {x \sqrt { a^2-x^2 }}{4} </math>
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|<math> \int x^2 \arcsin \frac {x }{ a}dx = \frac{x^3}{3}\arcsin \frac {x}{a} + \frac {\left( x^2+2a^2 \right) \sqrt { a^2-x^2 }}{9} </math>
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|-
+
−
|<math> \int \frac {\arcsin (x/a)}{x}dx = \frac {x}{a} + \frac {(a/x)^3}{2 \cdot 3 \cdot 3}+ \frac{1 \cdot 3(x/a)^5}{2 \cdot 4 \cdot 5 \cdot 5} + \frac {1 \cdot 3 \cdot 5 (x/a)^7}{2 \cdot 4 \cdot 6 \cdot 7 \cdot 7} + \cdot \cdot \cdot </math>
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|-
+
−
|<math> \int \frac {\arcsin (x/a)}{x^2}dx = -\frac{\arcsin (x/a)}{x} - \frac {1}{a} \ln { \left( \frac {a +\sqrt{a^2-x^2}}{x} \right)} </math>
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|-
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−
|<math> \int \left( \arcsin \frac{x}{a} \right)^2 dx = x \left( \arcsin \frac{x}{a} \right)^2-2x + 2 \sqrt{a^2-x^2} \arcsin \frac{x}{a} </math>
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|-
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|<math> \int \arccos \frac {x }{ a}dx = x \arccos \frac {x}{a} - \sqrt { a^2-x^2 } </math>
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|-
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|<math> \int x \arccos \frac {x }{ a}dx = \frac{x^2}{2}-\frac {a^2}{4} \arccos \frac {x}{a} - \frac {x \sqrt { a^2-x^2 }}{4} </math>
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|<math> \int x^2 \arccos \frac {x }{ a}dx = \frac{x^3}{3}\arccos \frac {x}{a} - \frac {\left( x^2+2a^2 \right) \sqrt { a^2-x^2 }}{9} </math>
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|-
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|<math> \int \frac {\arccos (x/a)}{x}dx = \frac {\pi}{2}\ln{x} - \int \frac {\arcsin (x/a)}{x} dx </math>
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|-
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|<math> \int \frac {\arccos (x/a)}{x^2}dx = -\frac{\arccos (x/a)}{x} + \frac {1}{a} \ln { \left( \frac {a +\sqrt{a^2-x^2}}{x} \right)} </math>
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|-
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−
|<math> \int \left( \arccos \frac{x}{a} \right)^2 dx = x \left( \arccos \frac{x}{a} \right)^2-2x - 2 \sqrt{a^2-x^2} \arccos \frac{x}{a} </math>
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|-
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−
|<math> \int \arctan \frac {x }{ a}dx = x \arctan \frac {x}{a} - \frac {a}{2} \ln {x^2+a^2} </math>
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|-
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−
|<math> \int x \arctan \frac {x }{ a}dx = \frac{1}{2}(x^2+a^2)\arctan \frac {x}{a} - \frac {ax}{2} </math>
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|<math> \int x^2 \arctan \frac {x }{ a}dx = \frac {x^3}{3}\arctan \frac{x}{a} - \frac {ax^2}{6} +\frac{a^3}{6} \ln \left (x^2+a^2) \right)  </math>
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|<math> \int \frac {\arctan (x/a)}{x}dx = \frac {x}{a} - \frac{(x/a)^3}{3^2} + \frac {(x/a)^5}{5^2} - \frac {(x/a)^7)}{7^2} + \cdot \cdot \cdot </math>
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−
|<math> \int \frac {\arctan (x/a)}{x^2}dx = -\frac{1}{x}\arctan \frac{x}{a} - \frac {a}{2} \ln { \left( \frac {\sqrt{a^2+x^2}}{x^2} \right)} </math>
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−
|<math> \int \arccot \frac {x }{ a}dx = x \arccot \frac {x}{a} + \frac {a}{2} \ln {x^2+a^2} </math>
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|-
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−
|<math> \int x \arccot \frac {x }{ a}dx = \frac{1}{2}(x^2+a^2)\arccot \frac {x}{a} + \frac {ax}{2} </math>
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−
|<math> \int x^2 \arccot \frac {x }{ a}dx = \frac {x^3}{3}\arccot \frac{x}{a} + \frac {ax^2}{6} - \frac{a^3}{6} \ln \left (x^2+a^2) \right)  </math>
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|<math> \int \frac {\arccot (x/a)}{x}dx = \frac {\pi}{2}\ln{x} - \int \frac{\arctan(x/a)}{x}dx </math>
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−
|<math> \int \frac {\arccot (x/a)}{x^2}dx = -\frac{\arccot(x/a)}{x} + \frac {1}{2a} \ln { \left( \frac {a^2+x^2}{x^2} \right)} </math>
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|-
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−
|<math> \int \arccos \frac {a}{x}dx=
+
−
\begin{cases}
+
−
x \arccos \frac{a}{x} - a \ln \left( x + \sqrt {x^2-a^2} \right) \qquad 0<\arccos \frac{a}{x}< \frac{\pi}{2} \\
+
−
x \arccos \frac{a}{x} + a \ln \left( x + \sqrt {x^2-a^2} \right) \qquad \frac {\pi}{2}<\arccos \frac{a}{x}< \pi \\
+
−
\end{cases} </math>
+
−
|-
+
−
|<math> \int x \arccos \frac {a}{x}dx=
+
−
\begin{cases}
+
−
\frac {x^2}{2} \arccos \frac{a}{x} - \frac {a \sqrt {x^2-a^2}}{2} \qquad 0<\arccos \frac{a}{x}< \frac{\pi}{2} \\
+
−
\frac {x^2}{2} \arccos \frac{a}{x} + \frac {a \sqrt {x^2-a^2}}{2} \qquad \frac {\pi}{2}<\arccos \frac{a}{x}< \pi \\
+
−
\end{cases} </math>
+
−
|-
+
−
|<math> \int x^2 \arccos \frac {a}{x}dx=
+
−
\begin{cases}
+
−
\frac {x^3}{3} \arccos \frac{a}{x} - \frac {ax \sqrt {x^2-a^2}}{6} - \frac {a^3}{6} \ln {\left(x+\sqrt {x^2-a^2}\right) }\qquad 0<\arccos \frac{a}{x}< \frac{\pi}{2} \\
+
−
\frac {x^3}{3} \arccos \frac{a}{x} + \frac {ax \sqrt {x^2-a^2}}{6} - \frac {a^3}{6} \ln {\left(x+\sqrt {x^2-a^2}\right) }\qquad \frac {\pi}{2}<\arccos \frac{a}{x}< \pi \\
+
−
\end{cases} </math>
+
−
|-
+
−
|<math> \int \frac {\arccos (x/a)}{x}dx = \frac {\pi}{2} + \frac {a}{x}+ \frac {(a/x)^3}{2 \cdot 3 \cdot 3}+ \frac{1 \cdot 3(x/a)^5}{2 \cdot 4 \cdot 5 \cdot 5} + \frac {1 \cdot 3 \cdot 5 (x/a)^7}{2 \cdot 4 \cdot 6 \cdot 7 \cdot 7} + \cdot \cdot \cdot </math>
+
−
|-
+
−
|<math> \int \frac {\arccos (x/a)}{x^2} dx=
+
−
\begin{cases}
+
−
-\frac{\arccos(a/x)}{x}+\frac{\sqrt{x^2-a^2}}{ax} \qquad 0<\arccos \frac{a}{x}< \frac{\pi}{2} \\
+
−
-\frac{\arccos(a/x)}{x}-\frac{\sqrt{x^2-a^2}}{ax} \qquad \frac{\pi}{2}< \arccos \frac{a}{x}<{\pi} \\
+
−
\end{cases} </math>
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|-
+
−
|<math> \int \arcsin \frac {a}{x}dx=
+
−
\begin{cases}
+
−
x \arcsin \frac{a}{x} + a \ln \left( x + \sqrt {x^2-a^2} \right) \qquad 0<\arcsin \frac{a}{x}< \frac{\pi}{2} \\
+
−
x \arcsin \frac{a}{x} + a \ln \left( x - \sqrt {x^2-a^2} \right) \qquad -\frac {\pi}{2}<\arcsin \frac{a}{x}< 0 \\
+
−
\end{cases} </math>
+
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|-
+
−
|<math> \int x \arcsin \frac {a}{x}dx=
+
−
\begin{cases}
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−
\frac {x^2}{2} \arcsin \frac{a}{x} + \frac {a \sqrt {x^2-a^2}}{2} \qquad 0<\arccos \frac{a}{x}< \frac{\pi}{2} \\
+
−
\frac {x^2}{2} \arcsin \frac{a}{x} + \frac {a \sqrt {x^2-a^2}}{2} \qquad -\frac {\pi}{2}<\arccos \frac{a}{x}< 0 \\
+
−
\end{cases} </math>
+
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|-
+
−
|<math> \int x^2 \arcsin \frac {a}{x}dx=
+
−
\begin{cases}
+
−
\frac {x^3}{3} \arcsin \frac{a}{x} + \frac {a \sqrt {x^2-a^2}}{2} + \frac {a^3}{6} \ln {\left(x+\sqrt {x^2-a^2}\right) }\qquad 0<\arcsin \frac{a}{x}< \frac{\pi}{2} \\
+
−
\frac {x^3}{3} \arcsin \frac{a}{x} - \frac {a \sqrt {x^2-a^2}}{2} - \frac {a^3}{6} \ln {\left(x+\sqrt {x^2-a^2}\right) }\qquad -\frac {\pi}{2}<\arccos \frac{a}{x}< 0 \\
+
−
\end{cases} </math>
+
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|-
+
−
|<math> \int \frac {\arcsin (x/a)}{x}dx = - \left( \frac {a}{x}+ \frac {(a/x)^3}{2 \cdot 3 \cdot 3}+ \frac{1 \cdot 3(x/a)^5}{2 \cdot 4 \cdot 5 \cdot 5} + \frac {1 \cdot 3 \cdot 5 (x/a)^7}{2 \cdot 4 \cdot 6 \cdot 7 \cdot 7} + \cdot \cdot \cdot \right)</math>
+
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|-
+
−
|<math> \int \frac {\arcsin (x/a)}{x^2} dx=
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\begin{cases}
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-\frac{\arcsin(a/x)}{x}-\frac{\sqrt{x^2-a^2}}{ax} \qquad 0<\arcsin \frac{a}{x}< \frac{\pi}{2} \\
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-\frac{\arcsin(a/x)}{x}+\frac{\sqrt{x^2-a^2}}{ax} \qquad -\frac{\pi}{2}< \arccos \frac{a}{x}<0 \\
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\end{cases} </math>
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|<math> \int x^m \arcsin \frac {x}{a}dx=\frac{x^{m+1}}{m+1}\arcsin \frac {x}{a} - \frac {1}{m+1} \int \frac{x^{m+1}}{\sqrt{a^2-x^2}}dx </math>
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|<math> \int x^m \arccos \frac {x}{a}dx=\frac{x^{m+1}}{m+1}\arccos \frac {x}{a} + \frac {1}{m+1} \int \frac{x^{m+1}}{\sqrt{a^2-x^2}}dx </math>
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|<math> \int x^m \arctan \frac {x}{a}dx=\frac{x^{m+1}}{m+1}\arcsin \frac {x}{a} - \frac {a}{m+1} \int \frac{x^{m+1}}{a^2+x^2}dx </math>
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|<math> \int x^m \arccot \frac {x}{a}dx=\frac{x^{m+1}}{m+1}\arcsin \frac {x}{a} + \frac {a}{m+1} \int \frac{x^{m+1}}{a^2+x^2}dx </math>
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|<math> \int x^m\arccos \frac {a}{x}dx=
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\begin{cases}
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\frac {x^{m+1} \arccos (a/x)}{m+1} - \frac {a}{m+1} \int \frac {x^mdx}{\sqrt{x^2-a^2}} \qquad 0<\arccos \frac{a}{x}< \frac{\pi}{2} \\
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\frac {x^{m+1} \arccos (a/x)}{m+1} + \frac {a}{m+1} \int \frac { x^mdx}{\sqrt {x^2-a^2} }\qquad \frac {\pi}{2}<\arccos \frac{a}{x}< \pi \\
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\end{cases} </math>
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|<math> \int x^m \arcsin \frac {a}{x}dx=
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\begin{cases}
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\frac {x^{m+1} \arcsin (a/x)}{m+1} + \frac {a}{m+1} \int \frac {x^m dx}{\sqrt{x^2-a^2}} \qquad 0<\arcsin \frac{a}{x}< \frac{\pi}{2} \\
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\frac {x^{m+1} \arcsin (a/x)}{m+1} - \frac {a}{m+1} \int \frac { x^m dx}{\sqrt {x^2-a^2}} \qquad -\frac {\pi}{2}<\arcsin \frac{a}{x}< 0 \\
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\end{cases} </math>
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! style="background-color: rgb(238, 238, 238); background-image: none; background-repeat: repeat; background-attachment: scroll; background-position: 0% 0%; -moz-background-size: auto auto; -moz-background-clip: -moz-initial; -moz-background-origin: -moz-initial; -moz-background-inline-policy: -moz-initial; font-size: 110%;" colspan="2" | 27 Integrals Component <math> e^{ax} </math>
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|<math> \int e^{ax}dx=\frac{e^{ax}}{a} </math>
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|<math> \int x e^{ax}dx=\frac{e^{ax}}{a}\left(x-\frac{1}{a} \right) </math>
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|<math> \int x^2 e^{ax}dx=\frac{e^{ax}}{a}\left(x^2-\frac{2x}{a}+\frac{2}{a^2}\right) </math>
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|<math> \int x^n e^{ax}dx=\frac{x^n e^{ax}}{a}-\frac{n}{a} \int x^{n-1} e^{ax}dx = \frac {e^{ax}}{a} \left( x^n- \frac{nx^{n-1}}{a}+\frac{n(n-1)x^{n-2}}{a^2}- \cdot \cdot \cdot \frac{(-1)^n n!}{a^n} \right ) \qquad \text{if n is a poaitive integer} </math>
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|<math> \int \frac {e^{ax}}{x}dx=\ln {x} + \frac {ax}{1 \cdot 1!} + \frac {(ax)^2}{2 \cdot 2!} + \frac {(ax)^3}{3 \cdot 3!} + \cdot \cdot \cdot </math>
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|<math> \int \frac {e^{ax}}{x^n}dx = \frac {-e^{ax}}{(n-1)x^{n-1}} + \frac {a}{n-1} \int \frac {e^{ax}}{x^{n-1}}dx </math>
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|<math> \int \frac {dx}{p+qe^{ax}}=\frac {x}{p}-\frac {1}{ap} \ln {\left (p+qe^{ax}\right)} </math>
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|<math> \int \frac {dx} {\left ( p+qe^{ax} \right) ^2}=\frac {x}{p^2}+\frac {1}{ap(p+qe^{ax})} -\frac{1}{ap^2}\ln {\left (p+qe^{ax}\right)} </math>
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|<math> \int \frac {dx}{pe^{ax}+qe^{-ax}}=
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\begin{cases}
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\frac {1}{a \sqrt{pq}} \arctan {\left ( \sqrt \frac {p}{q}e^{ax} \right)} \\
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\frac {1}{2a \sqrt{-pq}} \ln {\left( \frac{e^{ax}-\sqrt{-q/p}}{e^{ax}+\sqrt{-q/p}} \right)} \\
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\end{cases} </math>
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|<math> \int e^{ax} \sin bx dx = \frac {e^{ax}(a \sin bx-b \cos bx)}{a^2+b^2} </math>
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|<math> \int e^{ax} \cos bx dx = \frac {e^{ax}(a \cos bx-b \sin bx)}{a^2+b^2} </math>
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|<math> \int x e^{ax} \sin bx dx = \frac {x e^{ax}(a \sin bx - b \cos bx)}{a^2+b^2} - \frac {e^{ax} \left \{ (a^2-b^2)\sin bx -2ab \cos bx \right \} }{(a^2+b^2)^2} </math>
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|<math> \int x e^{ax} \cos bx dx = \frac {x e^{ax}(a \cos bx + b \sin bx)}{a^2+b^2} - \frac {e^{ax} \left \{ (a^2-b^2)\cos bx + 2ab \sin bx \right \} }{(a^2+b^2)^2} </math>
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|<math> \int e^{ax} \ln {x} dx = \frac {e^{ax} \ln {x}}{a}-\frac {1}{a} \int \frac {e^{ax}}{x}dx </math>
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|<math> \int e^{ax}\sin^n bx dx = \frac{e^{ax} \sin^{n-1}bx}{a^2+n^2 b^2}(a \sin bx -nb \cos bx) + \frac {n(n-1)b^2}{a^2+n^2 b^2} \int e^{ax} \sin^{n-2} bx dx </math>
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! style="background-color: rgb(238, 238, 238); background-image: none; background-repeat: repeat; background-attachment: scroll; background-position: 0% 0%; -moz-background-size: auto auto; -moz-background-clip: -moz-initial; -moz-background-origin: -moz-initial; -moz-background-inline-policy: -moz-initial; font-size: 110%;" colspan="2" | 24 Integrals of lnx
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| <math> \int\ln x dx=x\ln x-x </math>
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| <math> \int x\ln x dx=\dfrac{x^{2}}{2}(\ln x-\frac{1}{2}) </math>
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| <math> \int x^{m}\ln x dx=\dfrac{x^{m+1}}{m+1}(\ln x-\frac{1}{m+1}) </math>
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| <math> \int\dfrac{\ln x}{x} dx=\frac{1}{2}\ln^{2}x </math>
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| <math> \int\dfrac{\ln x}{x^{2}} dx=-\dfrac{\ln x}{x}-\dfrac{1}{x} </math>
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| <math> \int\ln^{2}x dx=x\ln^{2}x-2x\ln x+2x </math>
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| <math> \int\dfrac{\ln^{n}x}{x} dx=\dfrac{\ln^{n+1}x}{n+1} </math>
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| <math> \int\dfrac{dx}{x\ln x}=\ln(\ln x) </math>
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| <math> \int\dfrac{dx}{\ln x}=\ln(\ln x)+\ln x+\dfrac{\ln^{2}x}{2\cdot2!}+\dfrac{\ln^{3}x}{3\cdot3!}+\cdots </math>
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| <math> \int\dfrac{x^{m}dx}{\ln x}=\ln(\ln x)+(m+1)\ln x+\dfrac{(m+1)^{2}\ln^{2}x}{2\cdot2!}+\dfrac{(m+1)^{3}\ln^{3}x}{3\cdot3!}+\cdots </math>
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| <math> \int\ln^{n}x dx=x\ln^{n}x-n\int\ln^{n-1}x dx </math>
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| <math> \int x^{m}\ln^{n}x dx=\dfrac{x^{m+1}\ln^{n}x}{m+1}-\dfrac{n}{m+1}\int x^{m}\ln^{n-1}x dx </math>
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| <math> \int\ln(x^{2}+a^{2}) dx=x\ln(x^{2}+a^{2})-2x+2a\tan^{-1}\dfrac{x}{a} </math>
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| <math> \int\ln(x^{2}-a^{2}) dx=x\ln(x^{2}-a^{2})-2x+a\ln(\dfrac{x+a}{x-a}) </math>
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| <math> \int x^{m}\ln(x^{2}\pm a^{2}) dx=\dfrac{x^{m}\ln(x^{2}\pm a^{2})}{m+1}-\dfrac{2}{m+1}\int\dfrac{x^{m+2}}{x^{2}\pm a^{2}}dx </math>
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! style="background-color: rgb(238, 238, 238); background-image: none; background-repeat: repeat; background-attachment: scroll; background-position: 0% 0%; -moz-background-size: auto auto; -moz-background-clip: -moz-initial; -moz-background-origin: -moz-initial; -moz-background-inline-policy: -moz-initial; font-size: 110%;" colspan="2" | 25 Integrals of sh ax
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| <math> \int sh ax dx=\dfrac{ch ax}{a} </math>
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| <math> \int x sh ax dx=\dfrac{x ch ax}{a}-\dfrac{sh ax}{a^{2}} </math>
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| <math> \int x^{2} sh ax dx=(\dfrac{x^{2}}{a^{2}}+\dfrac{2}{a^{3}}) ch ax-\dfrac{2x}{a^{2}} sh ax </math>
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| <math> \int\dfrac{sh ax}{x} dx=ax+\dfrac{(ax)^{3}}{3\cdot3!}+\dfrac{(ax)^{5}}{5\cdot5!}+\cdots </math>
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| <math> \int\dfrac{sh ax}{x^{2}} dx=- \dfrac{sh ax}{x}+a \int\dfrac{ch ax}{x}dx </math>
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| <math> \int\dfrac{dx}{sh ax}=\dfrac{1}{a}\ln th\dfrac{ax}{2} </math>
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| <math> \int\dfrac{xdx}{sh ax}=\dfrac{1}{a^{2}}\{ax-\dfrac{(ax)^{3}}{18}+\dfrac{7(ax)^{5}}{1800}-\cdots+\dfrac{2(-1)^{n}(2^{2n}-1)B_{n}(ax)^{2n+1}}{(2n+1)!}\} </math>
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| <math> \int sh^{2} ax dx=\dfrac{sh ax ch ax}{2a}-\dfrac{x}{2} </math>
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| <math> \int x sh^{2} ax dx=\dfrac{x sh2ax}{4a}-\dfrac{ch2ax}{8a^{2}}-\dfrac{x^{2}}{4} </math>
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| <math> \int\dfrac{dx}{sh^{2} ax}=-\dfrac{coth ax}{a} </math>
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| <math> \int sh ax sh px dx=\dfrac{sh(a+p) x}{2(a+p)}-\dfrac{sh(a-p)x}{2(a-p)},
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p=\pm a </math>
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| <math> \int sh ax sin px dx=\dfrac{a ch ax sin px-p sh ax cos px}{a^{2}+p^{2}} </math>
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| <math> \int sh ax cos px dx=\dfrac{a ch ax cos px+p sh ax sin px}{a^{2}+p^{2}} </math>
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| <math> \int\dfrac{dx}{p+q sh ax}=\dfrac{1}{a\sqrt{p^{2}+q^{2}}}\ln(\dfrac{qe^{ax}+p-\sqrt{p^{2}+q^{2}}}{qe^{ax}+p+\sqrt{p^{2}+q^{2}}}) </math>
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| <math> \int\dfrac{dx}{(p+q sh ax)^{2}}=\dfrac{-q ch ax}{a(p^{2}+q^{2})(p+q sh ax)}+\dfrac{p}{p^{2}+q^{2}} \int\dfrac{dx}{p+q sh ax} </math>
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| <math> \int\dfrac{dx}{p^{2}+q^{2} sh^{2} ax}=\begin{cases}
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\dfrac{\dfrac{1}{ap\sqrt{q^{2}-p^{2}}}Arc tg\dfrac{\sqrt{q^{2}-p^{2}} th ax}{p}}{\dfrac{1}{2ap\sqrt{p^{2}-q^{2}}}\ln\biggl(\dfrac{p+\sqrt{p^{2}-q^{2}} th ax}{p-\sqrt{p^{2}-q^{2}} th ax}\biggl)} & .\end{cases}\dfrac{1}{a\sqrt{p^{2}+q^{2}}}\ln\biggl(\dfrac{qe^{ax}+p-\sqrt{p^{2}+q^{2}}}{qe^{ax}+p+\sqrt{p^{2}+q^{2}}}\biggl) </math>
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| <math> \int\dfrac{dx}{p^{2}-q^{2} sh^{2} ax}=\dfrac{1}{2ap\sqrt{p^{2}+q^{2}}}\ln(\dfrac{p+\sqrt{p^{2}+q^{2}} th ax}{p-\sqrt{p^{2}+q^{2}} th ax}) </math>
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| <math> \int x^{m} sh ax dx=\dfrac{x^{m} ch ax}{a}-\dfrac{m}{a}\int x^{m-1}ch ax dx </math>
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| <math> \int sh^{n} ax dx=\dfrac{sh^{n-1} ax ch ax}{an}-\dfrac{n-1}{n}\int sh^{n-2} ax dx </math>
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| <math> \int\dfrac{sh ax}{x^{n}} dx=\dfrac{-sh ax}{(n-1)x^{n-1}}+\dfrac{a}{n-1}\int\dfrac{ch ax}{x^{n-1}} dx </math>
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| <math> \int\dfrac{dx}{sh^{n} ax}=\dfrac{-ch ax}{a(n-1)sh^{n-1} ax}-\dfrac{n-2}{n-1}{\displaystyle \int}\dfrac{dx}{sh^{n-2} ax} </math>
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| <math> \int\dfrac{x}{sh^{n} ax} dx=\dfrac{-x ch ax}{a(n-1)sh^{n-1} ax}-\dfrac{1}{a^{2}(n-1)(n-2) sh^{n-2} ax}-\dfrac{n-2}{n-1}{\displaystyle \int}\dfrac{dx}{sh^{n-2} ax} </math>
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! style="background-color: rgb(238, 238, 238); background-image: none; background-repeat: repeat; background-attachment: scroll; background-position: 0% 0%; -moz-background-size: auto auto; -moz-background-clip: -moz-initial; -moz-background-origin: -moz-initial; -moz-background-inline-policy: -moz-initial; font-size: 110%;" colspan="2" | 26 Integrals of ch ax
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| <math> \int ch ax dx=\dfrac{sh ax}{a} </math>
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| <math> \int x ch ax dx=\dfrac{x sh ax}{a}-\dfrac{ch ax}{a^{2}} </math>
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| <math> \int x^{2} ch ax dx=(\dfrac{x^{2}}{a^{2}}+\dfrac{2}{a^{3}}) sh ax-\dfrac{2x}{a^{2}} ch ax </math>
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| <math> \int\dfrac{ch ax}{x} dx=\ln x+\dfrac{(ax)^{2}}{2\cdot2!}+\dfrac{(ax)^{4}}{4\cdot4!}+\cdots </math>
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| <math> \int\dfrac{ch ax}{x^{2}} dx=-\dfrac{ch ax}{x}+a \int\dfrac{sh ax}{x}dx </math>
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| <math> \int\dfrac{dx}{ch ax}=\dfrac{2}{a}Arc tg e^{ax} </math>
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| <math> \int\dfrac{xdx}{ch ax}=\dfrac{1}{a^{2}}\{\dfrac{(ax)^{2}}{2}-\dfrac{(ax)^{4}}{8}+\dfrac{5(ax)^{6}}{144}-\cdots+\dfrac{(-1)^{n}(2^{2n}-1)E_{n}(ax)^{2n+2}}{(2n+2)!}\} </math>
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| <math> \int ch^{2} ax dx=\dfrac{sh ax ch ax}{2a}+\dfrac{x}{2} </math>
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| <math> \int x ch^{2} ax dx=\dfrac{x sh2ax}{4a}-\dfrac{ch2ax}{8a^{2}}+\dfrac{x^{2}}{4} </math>
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| <math> \int\dfrac{dx}{ch^{2} ax}=\dfrac{th ax}{a} </math>
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| <math> \int ch ax ch px dx=\dfrac{sh(a-p) x}{2(a-p)}-\dfrac{sh(a+p)x}{2(a+p)} </math>
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| <math> \int ch ax sin px dx=\dfrac{a ch ax sin px-p sh ax cos px}{a^{2}+p^{2}} </math>
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| <math> \int ch ax cos px dx=\dfrac{a sh ax cos px+p ch ax sin px}{a^{2}+p^{2}} </math>
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| <math> \int\dfrac{dx}{ch ax+1}=\dfrac{1}{a} th\dfrac{ax}{2} </math>
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| <math> \int\dfrac{dx}{(ch ax-1)}=-\dfrac{1}{a} coth\dfrac{ax}{2} </math>
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| <math> \int\dfrac{xdx}{(ch ax+1)}=\dfrac{x}{a} th\dfrac{ax}{2}-\dfrac{2}{a^{2}}\ln ch\dfrac{ax}{2} </math>
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| <math> \int\dfrac{xdx}{(ch ax-1)}=-\dfrac{x}{a}coth\dfrac{ax}{2}+\dfrac{2}{a^{2}}\ln sh\dfrac{ax}{2} </math>
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| <math> \int\dfrac{dx}{(ch ax+1)^{2}}=\dfrac{1}{2a}th\dfrac{ax}{2}-\dfrac{1}{6a}th^{3}\dfrac{ax}{2} </math>
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| <math> \int\dfrac{dx}{(ch ax-1)^{2}}=\dfrac{1}{2a}coth\dfrac{ax}{2}-\dfrac{1}{6a}coth^{3}\dfrac{ax}{2} </math>
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| <math> \int\dfrac{dx}{p+q ch ax}=\begin{cases}
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\dfrac{\dfrac{2}{a\sqrt{q^{2}-p^{2}}}Arc tg\dfrac{q e^{ax}+p}{\sqrt{q^{2}-p^{2}}}}{\dfrac{1}{a\sqrt{p^{2}-q^{2}}}\ln\biggl(\dfrac{q e^{ax}+p-\sqrt{p^{2}-q^{2}}}{q e^{ax}+p+\sqrt{p^{2}-q^{2}}}\biggl)} & .\end{cases} </math>
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| <math> \int\dfrac{dx}{(p+q ch ax)^{2}}=\dfrac{q sh ax}{a(q^{2}-p^{2})(p+q ch ax)}-\dfrac{p}{q^{2}-p^{2}} \int\dfrac{dx}{p+q ch ax} </math>
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| <math> \int\dfrac{dx}{p^{2}-q^{2} ch^{2} ax}=\begin{cases}
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\dfrac{\dfrac{1}{2ap\sqrt{q^{2}-p^{2}}}\ln\biggl(\dfrac{p th ax+\sqrt{p^{2}-q^{2}}}{p th ax-\sqrt{p^{2}-q^{2}}}\biggl)}{\dfrac{1}{ap\sqrt{p^{2}-q^{2}}}-Arc tg\dfrac{p th ax}{\sqrt{q^{2}-p^{2}}}} & .\end{cases} </math>
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| <math> \int\dfrac{dx}{p^{2}+q^{2} ch ax}=\begin{cases}
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\dfrac{\dfrac{1}{2ap\sqrt{p^{2}+q^{2}}}\ln\biggl(\dfrac{p th ax+\sqrt{p^{2}+q^{2}}}{p th ax-\sqrt{p^{2}+q^{2}}}\biggl)}{\dfrac{1}{ap\sqrt{p^{2}+q^{2}}}-Arc tg\dfrac{p th ax}{\sqrt{p^{2}+q^{2}}}} & .\end{cases} </math>
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| <math> \int x^{m} ch ax dx=\dfrac{x^{m} sh ax}{a}-\dfrac{m}{a} \int x^{m-1}sh ax dx </math>
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| <math> \int ch^{n} ax dx=\dfrac{ch^{n-1} ax sh ax}{an}+\dfrac{n-1}{n} \int ch^{n-2} ax dx </math>
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| <math> \int\dfrac{ch ax}{x^{n}} dx=\dfrac{-ch ax}{(n-1)x^{n-1}}+\dfrac{a}{n-1} \int\dfrac{sh ax}{x^{n-1}} dx </math>
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| <math> \int\dfrac{dx}{ch^{n} ax}=\dfrac{-sh ax}{a(n-1)ch^{n-1} ax}+\dfrac{n-2}{n-1}{\displaystyle  \int}\dfrac{dx}{ch^{n-2} ax} </math>
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| <math> \int\dfrac{x}{ch^{n} ax} dx=\dfrac{-x sh ax}{a(n-1)ch^{n-1} ax}+\dfrac{1}{a^{2}(n-1)(n-2) ch^{n-2} ax}+\dfrac{n-2}{n-1}{\displaystyle  \int}\dfrac{dx}{ch^{n-2} ax} </math>
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! style="background-color: rgb(238, 238, 238); background-image: none; background-repeat: repeat; background-attachment: scroll; background-position: 0% 0%; -moz-background-size: auto auto; -moz-background-clip: -moz-initial; -moz-background-origin: -moz-initial; -moz-background-inline-policy: -moz-initial; font-size: 110%;" colspan="2" | 27 Integrals of th ax
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| <math> \int th ax dx=\dfrac{\ln ch ax}{a} </math>
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| <math> \int th^{2} ax dx=x-\dfrac{th ax}{a} </math>
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| <math> \int th^{3} ax dx=\dfrac{1}{a}\dfrac{\ln ch ax}{a}-\dfrac{th^{2} ax}{2a} </math>
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| <math> \int\dfrac{th^{n} ax}{ch^{2} ax} dx=\dfrac{th^{n+1} ax}{(n+1)a} </math>
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| <math> \int\dfrac{dx}{th ax ch^{2} ax} dx=\dfrac{1}{a}\ln th ax </math>
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| <math> \int\dfrac{dx}{th ax} dx=\dfrac{1}{a}\ln sh ax </math>
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| <math> \int x th ax dx=\dfrac{1}{a^{2}}\biggl\{\dfrac{(ax)^{3}}{3}-\dfrac{(ax)^{5}}{15}+\dfrac{2(ax)^{7}}{105}\cdots+\dfrac{(-1)^{n-1}2^{2n}(2^{2n}-1)B_{n}(ax)^{2n+1}}{(2n+1)|}\biggl\} </math>
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| <math> \int x th^{2} ax dx=\dfrac{x^{2}}{2}-\dfrac{x th ax}{a}+\dfrac{1}{a^{2}}\ln ch ax </math>
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| <math> \int\dfrac{th ax}{x} dx=\biggl\{ ax-\dfrac{(ax)^{3}}{9}+\dfrac{2(ax)^{5}}{75}-\cdots+\dfrac{(-1)^{n-1}2^{2n}(2^{2n}-1)B_{n}(ax)^{2n-1}}{(2n-1)(2n)!}\biggl\} </math>
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| <math> \int\dfrac{dx}{p+q th ax}=\dfrac{px}{p^{2}-q^{2}}-\dfrac{q}{a(p^{2}-q^{2})}\ln(q sh ax+p ch ax) </math>
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| <math> \int th^{n} ax dx=-\dfrac{th^{n+1} ax}{a(n-1)}+ \int th^{n-2} ax dx </math>
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! style="background-color: rgb(238, 238, 238); background-image: none; background-repeat: repeat; background-attachment: scroll; background-position: 0% 0%; -moz-background-size: auto auto; -moz-background-clip: -moz-initial; -moz-background-origin: -moz-initial; -moz-background-inline-policy: -moz-initial; font-size: 110%;" colspan="2" | 28 Integrals of coth ax
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| <math> \int coth ax dx=\dfrac{\ln sh ax}{a} </math>
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| <math> \int coth^{2} ax dx=x-\dfrac{coth ax}{a} </math>
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| <math> \int coth^{3} ax dx=\dfrac{1}{a}\dfrac{\ln sh ax}{a}-\dfrac{coth^{2} ax}{2a} </math>
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| <math> \int\dfrac{coth^{n} ax}{sh^{2} ax} dx=\dfrac{coth^{n+1} ax}{(n+1)a} </math>
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| <math> \int\dfrac{dx}{coth ax sh^{2} ax} dx=\dfrac{1}{a}\ln coth ax </math>
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| <math> \int\dfrac{dx}{coth ax} dx=\dfrac{1}{a}\ln ch ax </math>
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| <math> \int x coth ax dx=\dfrac{1}{a^{2}}\biggl\{ ax+\dfrac{(ax)^{3}}{9}-\dfrac{(ax)^{5}}{225}+\dfrac{2(ax)^{7}}{105}+\cdots\dfrac{(-1)^{n-1}2^{2n}B_{n}(ax)^{2n+1}}{(2n+1)|}\biggl\} </math>
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| <math> \int x coth^{2} ax dx=\dfrac{x^{2}}{2}-\dfrac{x coth ax}{a}+\dfrac{1}{a^{2}}\ln sh ax </math>
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| <math> \int\dfrac{coth ax}{x} dx=\biggl\{-\dfrac{1}{ax}+\dfrac{ax}{3}-\dfrac{(ax)^{3}}{135}+\cdots\dfrac{(-1)^{n-1}2^{2n}B_{n}(ax)^{2n-1}}{(2n-1)(2n)!}\biggl\} </math>
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| <math> \int\dfrac{dx}{p+q coth ax}=\dfrac{px}{p^{2}-q^{2}}-\dfrac{q}{a(p^{2}-q^{2})}\ln(p sh ax+q ch ax) </math>
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| <math> \int coth^{n} ax dx=-\dfrac{coth^{n-1} ax}{a(n-1)}+ \int coth^{n-2} ax dx </math>
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! style="background-color: rgb(238, 238, 238); background-image: none; background-repeat: repeat; background-attachment: scroll; background-position: 0% 0%; -moz-background-size: auto auto; -moz-background-clip: -moz-initial; -moz-background-origin: -moz-initial; -moz-background-inline-policy: -moz-initial; font-size: 110%;" colspan="2" | 29 Integrals of <math> \dfrac{1}{ch ax} </math>
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| <math> \int\dfrac{1}{ch ax}dx=\dfrac{2}{a}Arc tg e^{ax} </math>
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| <math> \int\dfrac{1}{ch^{2} ax}dx=\dfrac{th ax}{a} </math>
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| <math> \int\dfrac{1}{ch^{3} ax}dx=\dfrac{th ax}{2a ch ax}+\dfrac{1}{2a}Arc tg sh ax </math>
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| <math> \int\dfrac{th ax}{ch^{n} ax}dx=-\dfrac{1}{na ch^{n} ax} </math>
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| <math> \int ch ax dx=\dfrac{sh ax}{a} </math>
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| <math> \int\dfrac{xdx}{ch ax}=\dfrac{1}{a^{2}}\biggl\{\dfrac{(ax)^{3}}{2}-\dfrac{(ax)^{4}}{8}+\dfrac{5(ax)^{6}}{144}+\cdots+\dfrac{(-1)^{n}E_{n}(ax)^{2n+2}}{(2n+2)(2n)|}\biggl\} </math>
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| <math> \int\dfrac{xdx}{ch^{2} ax}=\dfrac{x th ax}{a}-\dfrac{1}{a^{2}}\ln ch ax </math>
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| <math> \int\dfrac{dx}{x sh ax}=\ln x-\dfrac{(ax)^{2}}{4}+\dfrac{5(ax)^{4}}{96}-\dfrac{6(ax)^{6}}{4320}+\cdots\dfrac{(-1)^{n}E_{n}(ax)^{2n}}{2n(2n)|}\biggl\} </math>
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| <math> \int\dfrac{dx}{q+\dfrac{p}{ch ax}}=\dfrac{x}{q}-\dfrac{p}{q} \int\dfrac{dx}{p+q ch ax} </math>
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| <math> \int\dfrac{1}{ch^{n} ax}dx=\dfrac{th ax}{a(n-1) ch^{n-2} ax}+\dfrac{(n-2)}{(n-1)} \int\dfrac{dx}{ch^{n-2} ax} </math>
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! style="background-color: rgb(238, 238, 238); background-image: none; background-repeat: repeat; background-attachment: scroll; background-position: 0% 0%; -moz-background-size: auto auto; -moz-background-clip: -moz-initial; -moz-background-origin: -moz-initial; -moz-background-inline-policy: -moz-initial; font-size: 110%;" colspan="2" | 30 Integrals of 1/sh ax
 
! style="background-color: rgb(238, 238, 238); background-image: none; background-repeat: repeat; background-attachment: scroll; background-position: 0% 0%; -moz-background-size: auto auto; -moz-background-clip: -moz-initial; -moz-background-origin: -moz-initial; -moz-background-inline-policy: -moz-initial; font-size: 110%;" colspan="2" | 30 Integrals of 1/sh ax

Revision as of 16:27, 26 February 2015


Collective Table of Formulas

Indefinite Integrals

click here for more formulas

Table of Indefinite Integrals
1 General Rules
$ \int a d x = a x $
$ \int a f ( x ) d x = a \int f ( x ) d x $
$ \int ( u \pm v \pm w \pm \cdot \cdot \cdot ) d x = \int u d x \pm \int v d x \pm \int w d x \pm \cdot \cdot \cdot $
$ \int u d v = u v - \int v d u $
$ \int f ( a x ) d x = \frac{1}{a} \int f ( u ) d u $
$ \int F \{ f ( x ) \} d x = \int F ( u ) \frac{dx}{du} d u = \int \frac{F ( u )}{f^{'} ( x )} d u \qquad u = f ( x ) $
$ \int u^n d u = \frac{u^{n+1}}{n+1} \qquad n \neq -1 $
$ \int \frac{d u}{u} = \ln u \ ( if \ u > 0 ) \ \text{or} \ln {-u} \ ( \text{if} \ u < 0 ) = \ln \left | u \right | $
$ \int e^u d u = e^u $
Click here for more general rules.
2 Important Transformations
$ \int F( a x + b) d x =\frac{1}{a} \int F( u) d u \qquad u = a x + b $
$ \int F( \sqrt {a x + b} ) d x =\frac{2}{a} \int u F( u) d u \qquad u = \sqrt {a x + b} $
$ \int F( \sqrt [n] {a x + b} ) d x = \frac{n}{a} \int u^{n-1} F( u) d u \qquad u = \sqrt [n] {a x + b} $
$ \int F( \sqrt {a^2 - x^2} ) d x =a \ \int F( a \cos u) \ \cos u \ d u \qquad x = a \sin u $
$ \int F( \sqrt {x^2 + a^2} ) d x =a \ \int F \left ( \frac {a}{\cos u} \right ) \frac {1}{\cos ^2 u} \ d u \qquad x = a \tan u $
$ \int F( \sqrt {x^2 - a^2} ) d x =a \ \int F \left ( a \tan u \right ) \frac {\tan u}{\cos u} \ d u \qquad x = \frac {a}{\cos u} $
$ \int F( e ^{a x}) d x = \frac {1}{a} \int \frac {F(u)}{u} \ d u \qquad u = e^{a x} $
$ \int F( \ln x ) d x = \int F(u)\ e^u \ d u \qquad u = \ln x $
$ \int F\left ( \arcsin \frac{x}{a} \right) d x = a \int F(u)\ \cos u \ d u \qquad u = \arcsin \frac {x}{a} $
$ \int F\left ( \sin x ,\cos x \right) d x = 2 \int F \left( \frac {2 u}{1 + u^2}, \frac {1 - u^2}{1+u^2} \right)\ \frac {d u}{1+ u^2} \qquad u = \tan \frac {x}{2} $
Integrals with ax +b
$ \int \frac {d x}{ ax + b} = \frac {1}{a} \ln (ax +b) $
$ \int \frac {x d x}{ ax + b} = \frac {x}{a} - \frac{b}{a^2} \ln (ax +b) $
$ \int \frac {x^2 d x}{ ax + b} = \frac {(ax+b)^2}{2a^3} - \frac {2b(ax+b) }{a^3} + \frac{b^2}{a^3} \ln (ax +b) $
$ \int \frac {d x}{\sqrt{a x +b}} = \frac {2\sqrt{ax+b}}{a} $
$ \int \frac {x d x}{\sqrt{a x + b}} = \frac {2(ax-2b)}{3a^2}\sqrt{ax+b} $
$ \int \frac {x^2 d x}{\sqrt{a x + b}} = \frac {2(3a^2x^2-4abx + 8b^2)}{15a^3}\sqrt{ax+b} $
Click here for more integrals with ax+b.

More

30 Integrals of 1/sh ax
$ \int\dfrac{1}{sh ax}dx=\dfrac{1}{a}\ln th\dfrac{ax}{2} $
$ \int\dfrac{1}{sh^{2} ax}dx=-\dfrac{coth ax}{a} $
$ \int\dfrac{1}{sh^{3} ax}dx=\dfrac{coth ax}{2a sh ax}+\dfrac{1}{2a}\ln th\dfrac{ax}{2} $
$ \int\dfrac{coth ax}{sh^{n} ax}dx=-\dfrac{1}{na sh^{n} ax} $
$ \int sh ax dx=\dfrac{ch ax}{a} $
$ \int\dfrac{xdx}{sh ax}=\dfrac{1}{a^{2}}\biggl\{ ax-\dfrac{(ax)^{3}}{18}+\dfrac{7(ax)^{5}}{1800}+\cdots+\dfrac{2(-1)^{n}(2^{2n-1}-1)B_{n}(ax)^{2n+1}}{(2n+1)|}\biggl\} $
$ \int\dfrac{xdx}{sh^{2} ax}=\dfrac{x coth ax}{a}-\dfrac{1}{a^{2}}\ln sh ax $
$ \int\dfrac{dx}{x sh ax}=-\dfrac{1}{ax}-\dfrac{ax}{6}+\dfrac{7(ax)^{3}}{1080}+\cdots\dfrac{(-1)^{n}(2^{2n-1}-1)B_{n}(ax)^{2n-1}}{(2n-1)(2n)|}\biggl\} $
$ \int\dfrac{dx}{q+\dfrac{p}{sh ax}}=\dfrac{x}{q}-\dfrac{p}{q} \int\dfrac{dx}{p+q sh ax} $
$ \int\dfrac{1}{sh^{n} ax}dx=\dfrac{coth ax}{a(n-1) sh^{n-2} ax}-\dfrac{(n-2)}{(n-1)} \int\dfrac{dx}{sh^{n-2} ax} $
31 Integrals of sh ax et ch ax
$ \int sh ax ch ax dx=\dfrac{sh^{2} ax}{2a} $
$ \int sh px ch qx dx=\dfrac{ch(p+q)x}{2(p+q)}+\dfrac{ch(p-q)x}{2(p-q)} $
$ \int sh^{n} ax ch ax dx=\dfrac{sh^{n+1} ax}{(n+1)a} $
$ \int ch^{n} ax sh ax dx=\dfrac{ch^{n+1} ax}{(n+1)a} $
$ \int sh^{2} ax ch^{2} ax dx=\dfrac{sh4ax}{32a}-\dfrac{x}{8} $
$ \int\dfrac{dx}{sh ax ch ax}=\dfrac{1}{a}\ln th ax $
$ \int\dfrac{dx}{sh^{2} ax ch ax}=-\dfrac{1}{a}Arc tg sh ax-\dfrac{1}{a sh ax} $
$ \int\dfrac{dx}{sh ax ch^{2} ax}=\dfrac{1}{a ch ax}+\dfrac{1}{a}\ln th\dfrac{ax}{2} $
$ \int\dfrac{dx}{sh^{2} ax ch^{2} ax}=-\dfrac{2 coth2ax}{a} $
$ \int\dfrac{sh^{2} ax dx}{ch ax}=-\dfrac{1}{a}Arc tg sh ax+\dfrac{sh ax}{a} $
$ \int\dfrac{ch^{2} ax dx}{sh ax}=\dfrac{1}{a}\ln th\dfrac{ax}{2}+\dfrac{ch ax}{a} $
$ \int\dfrac{dx}{sh ax(ch ax+1)}=\dfrac{1}{2a}\ln th\dfrac{ax}{2}+\dfrac{1}{2a(ch ax+1)} $
$ \int\dfrac{dx}{(sh ax+1) ch ax}=\dfrac{1}{2a}\ln\biggl(\dfrac{1+sh ax}{ch ax}\biggl)+\dfrac{1}{a}Arc tg e^{ax} $
$ \int\dfrac{dx}{sh ax(ch ax-1)}=-\dfrac{1}{2a}\ln th\dfrac{ax}{2}-\dfrac{1}{2a(ch ax-1)} $
32 Integrals of Hyperbolic Inverse functions : arg sh ax
$ \int\arg sh\dfrac{x}{a}dx=x\arg sh\dfrac{x}{a}-\sqrt{x^{2}+a^{2}} $
$ \int x\arg sh\dfrac{x}{a} dx=\biggl(\dfrac{x^{2}}{2}+\dfrac{a^{2}}{4}\biggl)\arg sh\dfrac{x}{a}-\dfrac{x\sqrt{x^{2}+a^{2}}}{4} $
$ \int x^{2}\arg sh\dfrac{x}{a} dx=\dfrac{x^{3}}{3}\arg sh\dfrac{x}{a}+\dfrac{(2a^{2-}x^{2})\sqrt{x^{2}+a^{2}}}{9} $
$ \int\dfrac{\arg sh\dfrac{x}{a}}{x}dx=\Biggl\{\begin{array}{c} \dfrac{x}{a}-\dfrac{(\dfrac{x}{a})^{3}}{2\cdot3\cdot3}+\dfrac{1\cdot3(\dfrac{x}{a})^{5}}{2\cdot4\cdot5\cdot5}-\dfrac{1\cdot3\cdot5(\dfrac{x}{a})^{7}}{2\cdot4\cdot6\cdot7\cdot7}+\cdots,|x|<a\\ \dfrac{\ln^{2}(\dfrac{2x}{a})}{2}-\dfrac{(\dfrac{a}{x})^{2}}{2\cdot2\cdot2}+\dfrac{1\cdot3(\dfrac{a}{x})^{4}}{2\cdot4\cdot4\cdot4}-\dfrac{1\cdot3\cdot5(\dfrac{a}{x})^{6}}{2\cdot4\cdot6\cdot6\cdot6}+\cdots, x>a\\ \dfrac{\ln^{2}(\dfrac{-2x}{a})}{2}+\dfrac{(\dfrac{a}{x})^{2}}{2\cdot2\cdot2}-\dfrac{1\cdot3(\dfrac{a}{x})^{4}}{2\cdot4\cdot4\cdot4}+\dfrac{1\cdot3\cdot5(\dfrac{a}{x})^{6}}{2\cdot4\cdot6\cdot6\cdot6}+\cdots, x<-a\end{array} $
$ \int\dfrac{\arg sh\dfrac{x}{a}}{x^{2}}dx=-\dfrac{\arg sh\dfrac{x}{a}}{x}-\dfrac{1}{a}\ln\Biggl(\dfrac{a+\sqrt{x^{2}+a^{2}}}{x}\Biggl) $
33 Integrals of Hyperbolic Inverse functions : arg ch ax
$ \int\arg ch\dfrac{x}{a}dx=\begin{cases} \dfrac{x\arg ch\dfrac{x}{a}-\sqrt{x^{2}-a^{2}}}{x\arg sh\dfrac{x}{a}+\sqrt{x^{2}-a^{2}}} & .\end{cases} $
$ \int x\arg ch\dfrac{x}{a} dx=\begin{cases} \dfrac{\frac{1}{4}(2x^{2}-a^{2})\arg ch\dfrac{x}{a}-\frac{1}{4}x\sqrt{x^{2}-a^{2}}}{\frac{1}{4}(2x^{2}-a^{2})\arg ch\dfrac{x}{a}+\frac{1}{4}x\sqrt{x^{2}-a^{2}}} & .\end{cases} $
$ \int x^{2}\arg ch\dfrac{x}{a} dx=\begin{cases} \dfrac{\frac{1}{3}x^{3}\arg ch\dfrac{x}{a}-\frac{1}{9}(x^{2}+2a^{2})\sqrt{x^{2}-a^{2}}}{\frac{1}{3}x^{3}\arg ch\dfrac{x}{a}+\frac{1}{9}(x^{2}+2a^{2})\sqrt{x^{2}-a^{2}}} & .\end{cases} $
$ \int\dfrac{\arg ch\dfrac{x}{a}}{x}dx=\pm\Biggl[\dfrac{\ln^{2}(\dfrac{2x}{a})}{2}+\dfrac{(\dfrac{a}{x})^{2}}{2\cdot2\cdot2}+\dfrac{1\cdot3(\dfrac{a}{x})^{4}}{2\cdot4\cdot4\cdot4}+\dfrac{1\cdot3\cdot5(\dfrac{a}{x})^{6}}{2\cdot4\cdot6\cdot6\cdot6}+\cdots,\Biggl] $
$ \int\dfrac{\arg ch\dfrac{x}{a}}{x^{2}}dx=-\dfrac{\arg ch\dfrac{x}{a}}{x}\mp\dfrac{1}{a}\ln\Biggl(\dfrac{a+\sqrt{x^{2}+a^{2}}}{x}\Biggl) $
34 Integrals of Hyperbolic Inverse functions : arg th ax
$ \int\arg th\dfrac{x}{a}dx=x\arg th\dfrac{x}{a}+\dfrac{a}{2}\ln(a^{2}-x^{2}) $
$ \int x\arg th\dfrac{x}{a} dx=\dfrac{ax}{2}+\frac{1}{2}(x^{2}-a^{2})\arg th\dfrac{x}{a} $
$ \int x^{2}\arg th\dfrac{x}{a} dx=\dfrac{ax^{2}}{6}+\frac{a^{3}}{6}\ln(a^{2}-x^{2})+\dfrac{x^{3}}{3}\arg th\dfrac{x}{a} $
$ \int\dfrac{\arg th\dfrac{x}{a}}{x}dx=\dfrac{x}{a}+\dfrac{(\dfrac{x}{a})^{3}}{3^{2}}+\dfrac{(\dfrac{x}{a})^{5}}{5^{2}}+\cdots $
$ \int\dfrac{\arg th\dfrac{x}{a}}{x^{2}}dx=-\dfrac{\arg th\dfrac{x}{a}}{x}+\dfrac{1}{2a}\ln\Biggl(\dfrac{x^{2}}{a^{2}-x^{2}}\Biggl) $
35 Integrals of Hyperbolic Inverse functions : arg coth ax
$ \int\arg coth\dfrac{x}{a}dx=x\arg coth x+\dfrac{a}{2}\ln(x^{2}-a^{2}) $
$ \int x\arg coth\dfrac{x}{a} dx=\dfrac{ax}{2}+\frac{1}{2}(x^{2}-a^{2})\arg coth\dfrac{x}{a} $
$ \int x^{2}\arg coth\dfrac{x}{a} dx=\dfrac{ax^{2}}{6}+\frac{a^{3}}{6}\ln(x^{2}-a^{2})+\dfrac{x^{3}}{3}\arg coth\dfrac{x}{a} $
$ \int\dfrac{\arg coth\dfrac{x}{a}}{x}dx=-\Biggl(\dfrac{a}{x}+\dfrac{(\dfrac{a}{x})^{3}}{3^{2}}+\dfrac{(\dfrac{a}{x})^{5}}{5^{2}}+\cdots\Biggl) $
$ \int\dfrac{\arg coth\dfrac{x}{a}}{x^{2}}dx=-\dfrac{\arg coth\dfrac{x}{a}}{x}+\dfrac{1}{2a}\ln\Biggl(\dfrac{x^{2}}{x^{2}-a^{2}}\Biggl) $
36 Integrals of a/x
$ \int\arg ch\dfrac{a}{x}dx=\begin{cases} \dfrac{x\arg ch\dfrac{a}{x}+\arcsin\dfrac{x}{a}}{x\arg ch\dfrac{a}{x}-\arcsin\dfrac{x}{a}} & .\end{cases} $
$ \int x\arg ch\dfrac{a}{x} dx=\begin{cases} \dfrac{\frac{1}{2}x^{2}\arg ch\dfrac{a}{x}-\dfrac{1}{2}a\sqrt{a^{2}-x^{2}}}{\frac{1}{2}x^{2}\arg ch\dfrac{a}{x}+\dfrac{1}{2}a\sqrt{a^{2}-x^{2}}} & .\end{cases} $
$ \int\dfrac{\arg ch\dfrac{a}{x}}{x}dx=\begin{cases} \dfrac{\dfrac{-\frac{1}{2}\ln(\dfrac{a}{x})\ln(\dfrac{4a}{x})}{2}-\dfrac{(\dfrac{x}{a})^{2}}{2\cdot2\cdot2}-\dfrac{1\cdot3(\dfrac{x}{a})^{4}}{2\cdot4\cdot4\cdot4}-\cdots}{\dfrac{\frac{1}{2}\ln(\dfrac{a}{x})\ln(\dfrac{4a}{x})}{2}+\dfrac{(\dfrac{x}{a})^{2}}{2\cdot2\cdot2}+\dfrac{1\cdot3(\dfrac{x}{a})^{4}}{2\cdot4\cdot4\cdot4}+\cdots} & .\end{cases} $
$ \int\arg sh\dfrac{a}{x}dx=x\arg sh\dfrac{a}{x}\pm\arg sh\dfrac{x}{a} $
$ \int x\arg sh\dfrac{a}{x} dx=\dfrac{x^{2}}{2}\arg sh\dfrac{a}{x}\pm\dfrac{1}{2}a\sqrt{a^{2}+x^{2}} $
$ \int\dfrac{\arg sh\dfrac{a}{x}}{x}dx=\Biggl\{\begin{array}{c} \dfrac{\frac{1}{2}\ln(\dfrac{x}{a})\ln(\dfrac{4a}{x})}{2}+\dfrac{(\dfrac{x}{a})^{2}}{2\cdot2\cdot2}-\dfrac{1\cdot3(\dfrac{x}{a})^{4}}{2\cdot4\cdot4\cdot4}+\cdots\\ \dfrac{\frac{1}{2}\ln(\dfrac{x}{a})\ln(\dfrac{4a}{x})}{2}-\dfrac{(\dfrac{x}{a})^{2}}{2\cdot2\cdot2}+\dfrac{1\cdot3(\dfrac{x}{a})^{4}}{2\cdot4\cdot4\cdot4}+\cdots\\ -\frac{a}{x}+\dfrac{(\dfrac{a}{x})^{3}}{2\cdot3\cdot3}-\dfrac{1\cdot3(\dfrac{a}{x})^{5}}{2\cdot4\cdot5\cdot5}+\cdots\end{array} $
$ \int x^{m}\arg sh\dfrac{x}{a} dx=\dfrac{x^{m+1}}{m+1}\arg sh\dfrac{x}{a}-\dfrac{1}{m+1} \int\dfrac{x^{m+1}}{\sqrt{x^{2}+a^{2}}}dx $
$ \int x^{m}\arg ch\dfrac{x}{a} dx=\begin{cases} \dfrac{\dfrac{x^{m+1}}{m+1}\arg ch\dfrac{x}{a}-\dfrac{1}{m+1} \int\dfrac{x^{m+1}}{\sqrt{x^{2}-a^{2}}}dx}{\dfrac{x^{m+1}}{m+1}\arg ch\dfrac{x}{a}+\dfrac{1}{m+1} \int\dfrac{x^{m+1}}{\sqrt{x^{2}-a^{2}}}dx} & .\end{cases} $
$ \int x^{m}\arg th\dfrac{x}{a} dx=\dfrac{x^{m+1}}{m+1}\arg th\dfrac{x}{a}-\dfrac{a}{m+1} \int\dfrac{x^{m+1}}{a^{2}-x^{2}}dx $
$ \int x^{m}\arg coth\dfrac{x}{a} dx=\dfrac{x^{m+1}}{m+1}\arg coth\dfrac{x}{a}-\dfrac{a}{m+1} \int\dfrac{x^{m+1}}{a^{2}-x^{2}}dx $
$ \int x^{m}\arg ch\dfrac{a}{x} dx=\begin{cases} \dfrac{\dfrac{x^{m+1}}{m+1}\arg ch\dfrac{a}{x}+\dfrac{a}{m+1} \int\dfrac{x^{m}}{\sqrt{a^{2}-x^{2}}}dx}{\dfrac{x^{m+1}}{m+1}\arg ch\dfrac{a}{x}-\dfrac{a}{m+1} \int\dfrac{x^{m}}{\sqrt{a^{2}-x^{2}}}dx} & .\end{cases} $
$ \int x^{m}\arg sh\dfrac{a}{x} dx=\dfrac{x^{m+1}}{m+1}\arg sh\dfrac{x}{a}\pm\dfrac{a}{m+1} \int\dfrac{x^{m}}{\sqrt{x^{2}+a^{2}}}dx $
Particular Integral, componant $ x^{2}-a^{2},x^{2}<a^{2} $
$ \int \dfrac{dx}{a^{2}-x^{2}} = \dfrac{1}{2a}\ln\left(\dfrac{a+x}{a-x}\right) \qquad o\grave{u}\qquad \dfrac{1}{a} Arg tg \dfrac{x}{a} $
$ \int \dfrac{xdx}{a^2 - x^2} = -\dfrac{1}{2} \ln\left({a^2-x^2}\right) $
$ \int \dfrac{x^2 dx}{a^2 - x^2} = -x + \dfrac{a}{2}\ln\left(\dfrac{a+x}{a-x}\right) $
$ \int \dfrac{x^3 dx}{a^2 - x^2} = -\dfrac{x^2}{2} - \dfrac{a^2}{2}\ln\left({a^2-x^2}\right) $
$ \int \dfrac{dx}{x\left(a^2-x^2\right)} = \dfrac{1}{2a^2}\ln\left(\dfrac{x^2}{a^2-x^2}\right) $
$ \int \dfrac{dx}{x^2\left(a^2-x^2\right)} = -\dfrac{1}{a^2x} + \dfrac{1}{2a^3}\ln\left(\dfrac{a+x}{a-x}\right) $
$ \int \dfrac{dx}{x^3\left(a^2-x^2\right)} = -\dfrac{1}{2a^2x^2} + \dfrac{1}{2a^4}\ln\left(\dfrac{x^2}{a^2-x^2}\right) $
$ \int \dfrac{dx}{\left(a^2-x^2\right)^2} = \dfrac{x}{2a^2\left(a^2-x^2\right)} + \dfrac{1}{4a^3}\ln\left(\dfrac{a+x}{a-x}\right) $
$ \int \dfrac{xdx}{\left(a^2-x^2\right)^2} = \dfrac{1}{2\left(a^2-x^2\right)} $
$ \int \dfrac{x^2dx}{\left(a^2-x^2\right)^2} = \dfrac{x}{2\left(a^2-x^2\right)} - \dfrac{1}{4a}\ln\left(\dfrac{a+x}{a-x}\right) $
$ \int \dfrac{x^3dx}{\left(a^2-x^2\right)^2} = \dfrac{a^2}{2\left(a^2-x^2\right)} - \dfrac{1}{2}\ln\left(a^2-x^2\right) $
$ \int \dfrac{dx}{x\left(a^2-x^2\right)^2} = \dfrac{1}{2a^2\left(a^2-x^2\right)} - \dfrac{1}{2a^4}\ln\left(\dfrac{x^2}{a^2-x^2}\right) $
$ \int \dfrac{dx}{x^2\left(a^2-x^2\right)^2} = \dfrac{-1}{a^4x} + \dfrac{x}{2a^4\left(a^2-x^2\right)} - \dfrac{3}{4a^5}\ln\left(\dfrac{a+x}{a-x}\right) $
$ \int \dfrac{dx}{x^3\left(a^2-x^2\right)^2} = \dfrac{-1}{2a^4x^2} + \dfrac{1}{2a^4\left(a^2-x^2\right)} + \dfrac{1}{a^6}\ln\left(\dfrac{x^2}{a^2-x^2}\right) $
$ \int \dfrac{dx}{\left(a^2-x^2\right)^n} = \dfrac{x}{2\left(n-1\right)a^2\left(a^2-x^2\right)^{n-1}} + \dfrac{2n-3}{\left(2n-2\right)a^2} \int\dfrac{dx}{\left(a^2-x^2\right)^{n-1}} $
$ \int \dfrac{xdx}{\left(a^2-x^2\right)^n} = \dfrac{1}{2\left(n-1\right)\left(a^2-x^2\right)^{n-1}} $
$ \int \dfrac{dx}{x\left(a^2-x^2\right)^n} = \dfrac{1}{2\left(n-1\right)\left(a^2-x^2\right)^{n-1}} + \dfrac{1}{a^2}\int\dfrac{dx}{x\left(a^2-x^2\right)^{n-1}} $
$ \int \dfrac{x^mdx}{\left(a^2-x^2\right)^n} = a^2\int\dfrac{x^{m-2}dx}{\left(a^2-x^2\right)^{n-1}} - \int\dfrac{x^{m-2}dx}{\left(a^2-x^2\right)^{n-1}} $
$ \int \dfrac{dx}{x^m\left(a^2-x^2\right)^n} = \dfrac{1}{a^2}\int\dfrac{dx}{x^m\left(a^2-x^2\right)^{n-1}} + \dfrac{1}{a^2}\int\dfrac{dx}{x^{m-2}\left(a^2-x^2\right)^n} $
Particular Integral, componant $ ax^2 + bx + c $
$ \int \dfrac{dx}{ax^2 + bx + c} = \begin{cases} \dfrac{2}{\sqrt{4ac-b^2}} \arctan \dfrac{2ax+b}{\sqrt{4ac-b^2}} \\ \dfrac{1}{\sqrt{b^2-4ac}} \ln\left(\dfrac{2ax+b-\sqrt{b^2-4ac}}{2ax+b+\sqrt{b^2-4ac}}\right) \end{cases} $
$ Si\quad b^2 = 4ac, ax^2 + bx + c = a\left(x+ b/2a\right)^2 et\quad on\quad peut\quad utiliser\quad les\quad r\acute{e}sultats\quad des\quad pages \quad60-61. $
$ \quad Si \quad b = 0 \quad utiliser \quad les \quad r\acute{e}sultats \quad de \quad la \quad page \quad 64. \quad Si \quad a \quad ou \quad c = 0,\quad utiliser \quad les $
$ \quad r\acute{e}sultats \quad des \quad pages \quad 60-61. $
$ \int \dfrac{xdx}{ax^2 + bx + c} = \dfrac{1}{2a} \ln\left(ax^2+bx+c\right) - \dfrac{b}{2a}\int \dfrac{dx}{ax^2 + bx + c} $
$ \int \dfrac{x^2dx}{ax^2 + bx + c} = \dfrac{x}{a} - \dfrac{b}{2a^2}\ln\left(ax^2+bx+c\right) + \dfrac{b^2-2ac}{2a^2} \dfrac{b}{2a}\int \dfrac{dx}{ax^2 + bx + c} $
$ \int \dfrac{x^{m}dx}{ax^2 + bx + c} = \dfrac{x^{m-1}}{\left(m-1\right)a} - \dfrac{c}{a}\int \dfrac{x^{m-2}dx}{ax^2 + bx + c} - \dfrac{b}{a}\int \dfrac{x{m-1}dx}{ax^2 + bx + c} $
$ \int \dfrac{dx}{x\left(ax^2 + bx + c\right)} = \dfrac{1}{2c} \ln\left(\dfrac{x^2}{ax^2+bx+c}\right) - \dfrac{b}{2c} \int \dfrac{dx}{ax^2+bx+c} $
$ \int \dfrac{dx}{x^2\left(ax^2 + bx + c\right)} = \dfrac{b}{2c^2} \ln\left(\dfrac{ax^2+bx+c}{x^2}\right) - \dfrac{1}{cx} + \dfrac{b^2-2ac}{2c^2} \int \dfrac{dx}{ax^2+bx+c} $
$ \int \dfrac{dx}{x^n\left(ax^2 + bx + c\right)} = - \dfrac{1}{\left(n-1\right)cx^{n-1}} - \dfrac{b}{c}\int \dfrac{dx}{x^{n-1}\left(ax^2+bx+c\right)} - \dfrac{a}{c}\int \dfrac{dx}{x^{n-2}\left(ax^2+bx+c\right)} $
$ \int \dfrac{dx}{\left(ax^2+bx+c \right)^2} = \dfrac{2ax+b}{\left(4ac-b^2 \right) \left(ax^2+bx+c \right)} + \dfrac{2a}{4ac-b^2} \int \dfrac{dx}{ax^2+bx+c} $
$ \int \dfrac{xdx}{\left(ax^2+bx+c^2 \right)^2} = - \dfrac{bx+2c}{\left(4ac-b^2\right)\left(ax^2+bx+c \right)} - \dfrac{b}{4ac-b^2} \int \dfrac{dx}{ax^2+bx+c} $
$ \int \dfrac{x^2 dx}{\left(ax^2+bx+c \right)^2} = \dfrac{\left(b^2-2ac\right)x+bc}{a\left(4ac-b^2\right)\left(ax^2+bx+c\right)} + \dfrac{2c}{4ac-b^2} \int \dfrac{dx}{ax^2+bx+c} $
$ \int \dfrac{x^{m}dx}{\left(ax^2+bx+c\right)^n} = - \dfrac{x^{m-1}}{\left(2n-m-1\right)a\left(ax^2+bx+c\right)^{n-1}} + \dfrac{\left(m-1\right)c}{\left(2n-m-1\right)a} \int \dfrac{x^{m-2}dx}{\left(ax^2+bx+c\right)^{n}} $
$ - \dfrac{\left(n-m\right)b}{\left(2n-m-1\right)a} \int \dfrac{x^{m-1}dx}{\left(ax^2+bx+c\right)^n} $
$ \int \dfrac{x^{2n-1}dx}{\left(ax^2+bx+c\right)^{n}} = \dfrac{1}{a} \int \dfrac{x^{2n-3}dx}{\left(ax^2+bx+c\right)^{n-1}} - \dfrac{c}{a} \int \dfrac{x^{2n-3}dx}{\left(ax^2+bx+c\right)^{n}} - \dfrac{b}{a} \int \dfrac{x^{2n-2}dx}{\left(ax^2+bx+c\right)^n} $
$ \int \dfrac{dx}{x\left(ax^2+bx+c\right)^2} = \dfrac{1}{2c\left(ax^2+bx+c\right)} - \dfrac{b}{2c} \int \dfrac{dx}{\left(ax^2+bx+c\right)^2} + \dfrac{1}{c} \int \dfrac{dx}{x\left(ac^2+bx+c\right)} $
$ \int \dfrac{dx}{x^2\left(ax^2+bx+c\right)^2} = -\dfrac{1}{cx\left(ax^2+bx+c\right)} - \dfrac{3a}{c} \int \dfrac{dx}{\left(ax^2+bx+c\right)^2} - \dfrac{2b}{c} \int \dfrac{dx}{x\left(ac^2+bx+c\right)^2} $
$ \int \dfrac{dx}{x^{m}\left(ax^2+bx+c\right)^{n}} = -\dfrac{1}{\left(m-1\right)cx^{m-1}\left(ax^2+bx+c\right)^{n-1}} - \dfrac{\left(m+2n-3\right)a}{\left(m-1\right)c} \int \dfrac{dx}{x^{m-2}\left(ax^2+bx+c\right)^{n}} $
$ + \dfrac{\left(m+n-2\right)b}{\left(m-1\right)c} \int \dfrac{dx}{x^{m-1}\left(ac^2+bx+c\right)^{n}} $
Particular Integral, componant $ \sqrt{ax^2 + bx + c} $
$ Dans\quad les\quad r\acute{e}sultats\quad suivants,\quad si\quad b^2 = 4ac, \sqrt{ax^2 + bx + c} = \sqrt{a}\left(x+ b/2a\right) et\quad on\quad peut\quad utiliser\quad $
$ les\quad r\acute{e}sultats\quad des\quad pages \quad60-61.\quad Si \quad b = 0 \quad utiliser \quad les \quad r\acute{e}sultats \quad de \quad la \quad page \quad 64. $
$ Si \quad a \quad ou \quad c = 0,\quad utiliser \quad les\quad r\acute{e}sultats \quad des \quad pages \quad 60-61. $
$ \int \dfrac{dx}{\sqrt{ax^2+bx+c}} = \begin{cases} \dfrac{1}{\sqrt{a}} \ln\left(2\sqrt{a}\sqrt{ax^2+bx+c}+ax+b\right)\\ -\dfrac{1}{\sqrt{-a}} \arcsin\left(\dfrac{2ax+b}{\sqrt{b^2-4ac}}\right)\quad ou\quad \dfrac{1}{\sqrt{a}} argsh\left(\dfrac{2ax+b}{\sqrt{4ac-b^2}}\right) \end{cases} $
$ \int \dfrac{xdx}{\sqrt{ax^2+bx+c}} = \dfrac{\sqrt{ax^2+bx+c}}{a} - \dfrac{b}{2a} \int \dfrac{dx}{\sqrt{ax^2+bx+c}} $
$ \int \dfrac{x^2dx}{\sqrt{ax^2+bx+c}} = \dfrac{2ax-3b}{4a^2}\sqrt{ax^2+bx+c}+\dfrac{3b^2-4ac}{8a^2}\int\dfrac{dx}{\sqrt{ax^2+bx+c}} $
$ \int \dfrac{dx}{x\sqrt{ax^2+bx+c}} = \begin{cases} -\dfrac{1}{\sqrt{c}}\ln\left(\dfrac{2\sqrt{c}\sqrt{ax^2+bx+c}+bx+2c}{x}\right)\\ -\dfrac{1}{\sqrt{-c}} \arcsin\left(\dfrac{bx+2c}{\left|x\right\vert\sqrt{b^2-4ac}}\right)\quad ou\quad -\dfrac{1}{\sqrt{c}}argsh\left(\dfrac{bx+2c}{\left|x\right\vert\sqrt{4ac-b^2}}\right)\end{cases} $
$ \int \dfrac{dx}{x^2\sqrt{ax^2+bx+c}} = -\dfrac{\sqrt{ax^2+bx+c}}{cx} - \dfrac{b}{2c}\int\dfrac{dx}{x\sqrt{ax^2+bx+c}} $
$ \int \sqrt{ax^2+bx+c} dx = \dfrac{\left(2ax+b\right)\sqrt{ax^2+bx+c}}{4a} + \dfrac{4ac-b^2}{8a}\int\dfrac{dx}{\sqrt{ax^2+bx+c}} $
$ \int x\sqrt{ax^2+bx+c} dx = \dfrac{\left(ax^2+bx+c\right)^{3/2}}{3a} - \dfrac{b\left(2ax+b\right)}{8a^2}\sqrt{ax^2+bx+c} - \dfrac{b\left(4ac-b^2\right)}{16a^2} \int \dfrac{dx}{\sqrt{ax^2+bx+c}} $
$ \int x^2\sqrt{ax^2+bx+c} dx = \dfrac{6ax-5b}{24a^2} \left(ax^2+bx+c\right)^{3/2} + \dfrac{5b^2-4ac}{16a^2}\int\sqrt{ax^2+bx+c} dx $
$ \int \dfrac{\sqrt{ax^2+bx+c}}{x} dx = \sqrt{ax^2+bx+c} + \dfrac{b}{2} \int\dfrac{dx}{\sqrt{ax^2+bx+c}} + c\int\dfrac{dx}{x\sqrt{ax^2+bx+c}} $
$ \int \dfrac{\sqrt{ax^2+bx+c}}{x^2} dx = -\dfrac{\sqrt{ax^2+bx+c}}{x} + a\int\dfrac{dx}{\sqrt{ax^2+bx+c}} + \dfrac{b}{2}\int\dfrac{dx}{x\sqrt{ax^2+bx+c}} $
$ \int \dfrac{dx}{\left(ax^2+bx+c\right)^{3/2}} = \dfrac{2\left(2ax+b\right)}{\left(4ac-b^2\right)\sqrt{ax^2+bx+c}} $
$ \int \dfrac{xdx}{\left(ax^2+bx+c\right)^{3/2}} = \dfrac{2\left(2bx+2c\right)}{\left(b^2-4ac\right)\sqrt{ax^2+bx+c}} $
$ \int \dfrac{x^2dx}{\left(ax^2+bx+c\right)^{3/2}} = \dfrac{\left(2b^2-4ac\right)x+2bc}{a\left(4ac-b^2\right)\sqrt{ax^2+bx+c}}+\dfrac{1}{a}\int\dfrac{dx}{\sqrt{ax^2+bx+c}} $
$ \int\dfrac{dx}{x\left(ax^2+bx+c\right)^{3/2}} = \dfrac{1}{c\sqrt{ax^2+bx+c}} + \dfrac{1}{c}\int\dfrac{dx}{x\sqrt{ax^2+x+c}} - \dfrac{b}{2c}\int\dfrac{dx}{\left(ax^2+bx+c\right)^{3/2}} $
$ \int\dfrac{dx}{x^2\left(ax^2+bx+c\right)^{3/2}} = -\dfrac{ax^2+2bx+c}{c^2x\sqrt{ax^2+bx+c}} + \dfrac{v^2-2ac}{2c^2}\int\dfrac{dx}{\left(ax^2+x+c\right)^{3/2}} - \dfrac{3b}{2c^2}\int\dfrac{dx}{x\sqrt{ax^2+bx+c}} $
$ \int\left(ax^2+bx+c\right)^{n+1/2} dx = \dfrac{\left(2ax+b\right)\left(ax^2+bx+c\right)^{n+1/2}}{4a\left(n+1\right)} + \dfrac{\left(2n+1\right)\left(4ac-b^2\right)}{8a\left(n+1\right)}\int\left(ax^2+bx+c\right)^{n-1/2}dx $
$ \int x\left(ax^2+bx+c\right)^{n+1/2}dx = \dfrac{\left(ax^2+bx+c\right)^{n+3/2}}{a\left(2n+3\right)} - \dfrac{b}{2a}\int \left((ax^2+bx+c\right)^{n+1/2} dx $
$ \int \dfrac{dx}{\left(ax^2+bx+c\right)^{n+1/2}} = \dfrac{2\left(2ax+b\right)}{\left(2n-1\right)\left(4ac-b^2\right)\left(ax^2+bx+c\right)^{n-1/2}} + \dfrac{8a\left(n-1\right)}{\left(2n-1\right)\left(4ac-b^2\right)} \int \dfrac{dx}{\left(ax^2+bx+c\right)^{n-1/2}} $
$ \int \dfrac{dx}{x\left(ax^2+bx+c\right)^{n+1/2}} = \dfrac{1}{\left(2n-1\right)c\left(ax^2+bx+c\right)^{n-1/2}} + \dfrac{1}{c} \int \dfrac{dx}{x\left(ax^2+bx+c\right)^{n-1/2}} - \dfrac{b}{2c} \int \dfrac{dx}{\left(ax^2+bx+c\right)^{n+1/2}} $
Particular Integral, componant $ x^3+a^3 $
$ \int \dfrac{dx}{x^3+a^3} = \dfrac{1}{6a^2} \ln \dfrac{\left(x+a\right)^2}{x^2-ax+a^2} + \dfrac{1}{a^2\sqrt{3}} \arctan \dfrac{2x-a}{a\sqrt{3}} $
$ \dfrac{xdx}{x^3+a^3} = \dfrac{1}{6a}\ln \dfrac{x^2-ax+a^2}{\left(x+a\right)^2} + \dfrac{1}{a\sqrt{3}} \arctan \dfrac{2x-a}{a\sqrt{3}} $
$ \int \dfrac{x^2dx}{x^3+a^3} = \dfrac{1}{3} \ln\left(x^3+a^3\right) $
$ \int \dfrac{dx}{x\left(x^3+a^3\right)} = \dfrac{1}{3a^3} \ln \left(\dfrac{x^3}{x^3+a^3}\right) $
$ \int \dfrac{dx}{x^2\left(x^3+a^3\right)} = -\dfrac{1}{a^3x} - \dfrac{1}{6a^4} \ln \dfrac{x^2-ax+a^2}{\left(x+a\right)} - \dfrac{1}{a^4\sqrt{3}} \arctan \dfrac{2x-a}{a\sqrt{3}} $
$ \int \dfrac{dx}{\left(x^3+a^3\right)^2} = \dfrac{x}{3a^3\left(x^3+a^3\right)} + \dfrac{1}{9a^5} \ln\dfrac{\left(x+a\right)^2}{x^2-ax+a^2} + \dfrac{2}{3a^5\sqrt{3}}\arctan\dfrac{2x-a}{a\sqrt{3}} $
$ \int \dfrac{xdx}{\left(x^3+a^3\right)^2} = \dfrac{x^2}{3a^3\left(x^3+a^3\right)} + \dfrac{1}{18a^4} \ln \dfrac{x^2-ax+a^2}{\left(x+a\right)^2} + \dfrac{1}{3a^4\sqrt{3}} \arctan \dfrac{2x-a}{a\sqrt{3}} $
$ \int \dfrac{x^2dx}{\left(x^3+a^3\right)} = -\dfrac{1}{3\left(x^3+a^3\right)} $
$ \int \dfrac{dx}{x\left(x^3+a^3\right)} = \dfrac{1}{3a^3\left(x^3+a^3\right)} + \dfrac{1}{3a^6}\ln\left(\dfrac{x^3}{x^3+a^3}\right) $
$ \int \dfrac{dx}{x^2\left(x^3+a^3\right)} = -\dfrac{1}{a^6x} - \dfrac{x^2}{3a^6\left(x^3+a^3\right)} - \dfrac{4}{3a^6}\int \dfrac{xdx}{x^3+a^3} $
$ \int \dfrac{x^{m}dx}{x^3+a^3} = \dfrac{x^{m-2}}{m-2} - a^3 \int \dfrac{x^{m-3}dx}{x^3+a^3} $
$ \int \dfrac{dx}{x^{n}\left(x^3+a^3\right)} = \dfrac{-1}{a^3\left(n-1\right)x^{n-1}} - \dfrac{1}{a^3} \int \dfrac{dx}{x^{n-3}\left(x^3+a^3\right)} $
Particular Integral, componant $ x^4 \pm a^4 $
$ \int \dfrac{dx}{x^4+a^4} = \dfrac{1}{4a^3\sqrt{2}}\ln\left(\dfrac{x^2+ax\sqrt{2}+a^2}{x^2-ax\sqrt{2}+a^2}\right) - \dfrac{1}{2a^3\sqrt{2}}\arctan \dfrac{ax\sqrt{2}}{x^2-a^2} $
$ \int \dfrac{xdx}{x^4+a^4} = \dfrac{1}{2a^2} \arctan \dfrac{x^2}{a^2} $
$ \int \dfrac{x^2da}{x^4+a^4} = \dfrac{1}{4a\sqrt{2}} \ln\left(\dfrac{x^2-ax\sqrt{2}+a^2}{x^2+ax\sqrt{2}+a^2}\right) = \dfrac{1}{2a\sqrt{2}} \arctan \dfrac{ax\sqrt{2}}{x^2-a^2} $
$ \int \dfrac{x^3dx}{x^4+a^4} = \dfrac{1}{4} \ln \left(x^4+a^4\right) $
$ \int \dfrac{dx}{x\left(x^4+a^4\right)} = \dfrac{1}{4a^4}\ln\left(\dfrac{x^4}{x^4+a^4}\right) $
$ \int \dfrac{dx}{x^2\left(x^4+a^4\right)} = -\dfrac{1}{a^4x} - \dfrac{1}{4a^5\sqrt{2}}\ln\left(\dfrac{x^2-ax\sqrt{2}+a^2}{x^2+ax\sqrt{2}+a^2}\right) + \dfrac{1}{2a^5\sqrt{2}}\arctan\dfrac{ax\sqrt{2}}{x^2-a^2} $
$ \int \dfrac{dx}{x^3\left(x^4+a^4\right)} = -\dfrac{1}{2a^4x^2} - \dfrac{1}{a^6} \arctan\dfrac{x^2}{a^2} $
$ \int \dfrac{dx}{x^4-a^4} = \dfrac{1}{4a^3}\ln\left(\dfrac{x-a}{x+a}\right) - \dfrac{1}{2a^3}\arctan\dfrac{x}{a} $
$ \int \dfrac{xdx}{x^4-a^4} = \dfrac{1}{4a^2}\ln\left(\dfrac{x^2-a^2}{x^2+a^2}\right) $
$ \int \dfrac{x^2dx}{x^4-a^4} = \dfrac{1}{4a}\ln\left(\dfrac{x-a}{x+a}\right) + \dfrac{1}{2a}\arctan\dfrac{x}{a} $
$ \int \dfrac{x^3dx}{x^4-a^4} = \dfrac{1}{4} \ln\left(x^4-a^4\right) $
$ \int \dfrac{dx}{x\left(x^3-a^4\right)} = \dfrac{1}{4a^4}\ln\left(\dfrac{x^4-a^4}{x^4}\right) $
$ \int \dfrac{dx}{x^2\left(x^4-a^4\right)} = \dfrac{1}{a^4x} + \dfrac{1}{4a^5}\ln\left(\dfrac{x-a}{x+a}\right) + \dfrac{1}{2a^5}\arctan\dfrac{x}{a} $
$ \int \dfrac{dx}{x^3\left(x^4-a^4\right)} = \dfrac{1}{2a^4x^2} + \dfrac{1}{4a^6}\ln\left(\dfrac{x^2-a^2}{x^2+a^2}\right) $
Particular Integral, componant $ x^{n} + a^{n} $
$ \int \dfrac{dx}{x\left(x^n+a^n\right)} = \dfrac{1}{na^{n}}\ln\dfrac{x^{n}}{x^{n}+a^{n}} $
$ \int \dfrac{x^{n-1}}{x^{n}+a^{n}} = \dfrac{1}{n}\ln\left(x^{n}+a^{n}\right) $
$ \int \dfrac{x^{m}}{\left(x^{n}+a^{n}\right)^{r}} = \int \dfrac{x^{m-n}dx}{\left(x^{n}+a^{n}\right)^{r-1}} - a^{n}\int\dfrac{x^{m-n}dx}{\left(x^{n}+a^{n}\right)^{r}} $
$ \int \dfrac{dx}{x^{m}\left(x^{n}+a^{n}\right)^{r}} = \dfrac{1}{a^{n}}\int\dfrac{dx}{x^{m}\left(x^{n}+a{n}\right)^{r-1}} - \dfrac{1}{a^{n}}\int\dfrac{dx}{x^{m-n}\left(x^{n}+a^{n}\right)^{r}} $
$ \int \dfrac{dx}{x\sqrt{x^{n}+a^{n}}} = \dfrac{1}{n\sqrt{a^{n}}}\ln\left(\dfrac{\sqrt{x^{n}+a^{n}}-\sqrt{a^{n}}}{\sqrt{x^{n}+a^{n}}+\sqrt{a^{n}}}\right) $
$ \int \dfrac{dx}{x\left(x^{n}-a^{n}\right)} = \dfrac{1}{na^{n}}\ln\left(\dfrac{x^{n}-a^{n}}{x^{n}}\right) $
$ \int \dfrac{x^{n-1}dx}{x^{n}-a^{n}} = \dfrac{1}{n}\ln\left(x^{n}-a^{n}\right) $
$ \int \dfrac{x^{m}dx}{\left(x^{n}-a^{n}\right)^{r}} = a^{n}\int\dfrac{x^{m-n}dx}{\left(x^{n}-a^{n}\right)^{r}} + \int\dfrac{x^{m-n}dx}{\left(x^{n}-a^{n}\right)^{r-1}} $
$ \int\dfrac{dx}{x^{m}\left(x^{n}-a^{n}\right)^{r}} = \dfrac{1}{a^{n}}\int\dfrac{dx}{x^{m-n}\left(x^{n}-a^{n}\right)^{r}}-\dfrac{1}{a^{n}}\int\dfrac{dx}{x^{m}\left(x^{n}-a^{n}\right)^{r-1}} $
$ \int\dfrac{dx}{x\sqrt{x^{n}-a^{n}}} = \dfrac{2}{n\sqrt{a^{n}}}\arccos\sqrt{\dfrac{a^{n}}{x^{n}}} $
$ \int \dfrac{x^{p-1}dx}{x^{2m}+a^{2m}} = \dfrac{1}{ma^{2m-p}} \sum_{k=1}^m \sin\dfrac{\left(2k-1\right)p\pi}{2m}\arctan\left(\dfrac{x+a\cos\left[\left(2k-1\right)\pi /2m\right]}{a\sin\left[\left(2k-1\right)\pi /2m\right]}\right) - \dfrac{1}{2ma^{2m-p}} \sum_{k=1}^m \cos\dfrac{\left(2k-1\right)p\pi}{2m}\ln\left(x^2+2ax\cos\dfrac{\left(2k-1\right)\pi}{2m} + a^2\right) $
$ \int \dfrac{x^{p-1}dx}{x^{2m}-a^{2m}} = \dfrac{1}{2ma^{2m-p}} \sum_{k=1}^{m-1} \cos\dfrac{kp\pi}{m}\ln\left(x^2 - 2ax\cos\dfrac{k\pi}{m} + a^2\right) - \dfrac{1}{ma^{2m-p}} \sum_{k=1}^{m-1} \sin \dfrac{kp\pi}{m} \arctan\left(\dfrac{x-a\cos\left(k\pi /m\right)}{a\sin\left(k\pi/m\right)}\right) + \dfrac{1}{2ma^{2m-p}}\left\{\ln\left(x-a\right)+\left(-1\right)^{p}\ln\left(x+a\right)\right\} $
$ \int \dfrac{x^{p-1}dx}{x^{2m+1}+a^{2m+1}} = \dfrac{2\left(-1\right)^{p-1}}{\left(2m+1\right)a^{2m-p+1}}\sum_{k=1}^m\sin\dfrac{2kp\pi}{2m+1}\arctan\left(\dfrac{x+a\cos\left[2k\pi/ \left(2m+1\right)\right]}{a\sin\left[2k\pi/ \left(2m+1\right)\right]}\right)-\dfrac{\left(-1\right)^{p-1}}{\left(2m+1\right)a^{2m-p+1}}\sum_{k=1}^m \cos\dfrac{2kp\pi}{2m+1}\ln\left(x^2+2ax\cos\dfrac{2k\pi}{2m+1}+a^2\right) + \dfrac{\left(-1\right)^{p-1}}{\left(2m+1\right)a^{2m-p+1}} $
$ \int \dfrac{x^{p-1}dx}{x^{2m+1}-a^{2m+1}} = \dfrac{-2}{\left(2m+1\right)a^{2m-p+1}} \sum_{k=1}^m \sin \dfrac{2kp\pi}{2m+1}\arctan\left(\dfrac{x-a\cos\left[2k\pi/\left(2m+1\right)\right]}{a\sin\left[2k\pi/\left(2m+1\right)\right]}\right) + \dfrac{1}{\left(2m+1\right)a^{2m-p+1}}\sum_{k=1}^m \cos\dfrac{2kp\pi}{2m+1}\ln\left(x^2-2ax\cos\dfrac{2k\pi}{2m+1}+a^2\right) + \dfrac{\ln\left(x-a\right)}{\left(2m+1\right)a^{2m-p+1}} $

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Ph.D. 2007, working on developing cool imaging technologies for digital cameras, camera phones, and video surveillance cameras.

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