| Table of Infinite Integrals | |
|---|---|
| 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 ) \ or \ln {-u} \ ( if \ u < 0 ) = \ln \left | u \right | $ | |
| $ \int e^u d u = e^u $ | |
| $ \int a^u d u = \int e^{u \ln a} d u = \frac{e^{u \ln a}}{\ln a} = \frac{a^u}{\ln a} \qquad a > 0 \ and \ a \neq 1 $ | |
| $ \int \sin u \ d u = - \cos u $ | |
| $ \int \cos u \ d u = \sin u $ | |
| $ \int \tan u \ d u = - \ln {\cos u} $ | |
| $ \int \cot u \ d u = \ln {\sin u} $ | |
| $ \int \frac{d u}{\cos u} = \ln { \left ( \frac{1}{\cos u} + \tan u \right )} = \ln{\tan {\left ( \frac{u}{2}+\frac{\pi}{4}\right )}} $ | |
| $ \int \frac{d u}{\sin u} = \ln { \left ( \frac{1}{\sin u} - \cot u \right )} = \ln{\tan { \frac{u}{2}}} $ | |
| $ \int \frac{d u}{\cos ^2 u} = \tan u $ | |
| $ \int \frac{d u}{\sin ^2 u} = - \cot u $ | |
| $ \int \tan ^2 u \ d u = \tan u - u $ | |
| $ \int \cot ^2 u \ d u = - \cot u - u $ | |
| $ \int \sin ^2 u \ d u= \frac{u}{2} - \frac{\sin {2 u}}{4} = \frac{1}{2}\left( u - \sin u \cos u \right ) $ | |
| $ \int \frac {1}{\cos u} \tan u \ d u = \frac{1}{\cos u} $ | |
| $ \int \frac {1}{\sin u} \cot u \ d u = - \frac{1}{\sin u} $ | |
| $ \int \operatorname{sh}\,u \ d u = \operatorname{ch}\,u $ | |
| $ \int \operatorname{ch}\,u \ d u = \operatorname{sh}\,u $ | |
| $ \int \operatorname{th}\,u \ d u = \ln \operatorname{ch}\,u $ | |
| $ \int \operatorname{coth}\,u \ d u = \ln \operatorname{sh}\,u $ | |
| $ \int \frac {1}{\operatorname{ch}\ u} \ d u = \arcsin{\left ( \operatorname{th}\,u \right )} \qquad or \ 2 arc \ th \ e^u $ | |
| $ \int \frac {1}{\operatorname{sh}\ u} \ d u = \ln \operatorname{th}\,\frac{2}{2} \qquad or \ - \operatorname{Arg coth} \ e^u $ | |
| $ \int \frac {1}{\operatorname{ch^2}\ u} \ d u = \operatorname{th}\,u $ | |
| $ \int \frac {1}{\operatorname{sh^2}\ u} \ d u = - \operatorname{coth}\,u $ | |
| $ \int \operatorname{th^2}\ u \ d u = u - \operatorname{th}\,u $ | |
| $ \int \operatorname{coth^2}\ u \ d u = u - \operatorname{coth}\,u $ | |
| $ \int \operatorname{sh^2}\ u \ d u = \frac {\operatorname{sh}\,{2 u}}{4} - \frac{u}{2}=\frac{1}{2}\left ( \operatorname{sh}\,u \ \operatorname{ch}\,u - u \right ) $ | |
| $ \int \operatorname{ch^2}\ u \ d u = \frac {\operatorname{sh}\,{2 u}}{4} + \frac{u}{2}=\frac{1}{2}\left ( \operatorname{sh}\,u \ \operatorname{ch}\,u + u \right ) $ | |
| $ \int \frac{\operatorname th \ u}{\operatorname ch \ u} \ d u = - \frac {1}{\operatorname ch \, u } $ | |
| $ \int \frac{\operatorname coth \ u}{\operatorname sh \ u} \ d u = - \frac {1}{\operatorname sh \, u } $ | |
| $ \int \frac{d u}{u^2 + a^2} = \frac {1}{a}\arctan \frac{u}{a} $ | |
| $ \int \frac{d u}{u^2 - a^2} = \frac {1}{2 a}\ln \left ( \frac{u-a}{u+a} \right ) = -\frac{1}{a} \operatorname{argcoth} \ \frac{u}{a} \qquad u^2 > a^2 $ | |
| $ \int \frac{d u}{a^2 - u^2} = \frac {1}{2 a}\ln \left ( \frac{a+u}{a-u} \right ) = \frac{1}{a} \operatorname{argth}\ \frac{u}{a} \qquad u^2 < a^2 $ | |
| $ \int \frac{d u}{\sqrt{a^2 - u^2}} = \arcsin \frac{u}{a} $ | |
| $ \int \frac{d u}{\sqrt{u^2 + a^2}} = \ln { \left ( u + \sqrt {u^2+a^2} \right ) } \qquad or \ \operatorname{argth} \ \frac{u}{a} $ | |
| $ \int \frac{d u}{\sqrt{u^2 - a^2}} = \ln { \left ( u + \sqrt {u^2-a^2} \right ) } $ | |
| $ \int \frac{d u}{u \sqrt{u^2 - a^2}} = \frac {1}{a} \arccos \left | \frac{a}{u} \right | $ | |
| $ \int \frac{d u}{u \sqrt{u^2 + a^2}} = - \frac {1}{a} \ln \left ( \frac{a + \sqrt{u^2 + a^2}}{u} \right ) $ | |
| $ \int \frac{d u}{u \sqrt{a^2 - u^2}} = - \frac {1}{a} \ln \left ( \frac{a + \sqrt{a^2 - u^2}}{u} \right ) $ | |
| $ \int f^{(n)} \ g d x =f^{(n-1)} \ g - f^{(n-2)} \ g' + f^{(n-3)} \ g'' - \cdot \cdot \cdot \ (-1)^n \int fg^{(n)} d x $ | |
| 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} $ | |
| Particular Integral, component 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 {x^3 d x}{ ax + b} = \frac {(ax+b)^3}{3a^4} - \frac {3b(ax+b)^2 }{2a^4} + \frac{3b^2(ax+b)}{a^4} - frac{b^3}{a^3}\ln (ax +b) $ | |
| $ \int \frac {d x}{ x(ax + b)} = \frac {1}{b} \ln \left ( \frac {x}{ax +b} \right) $ | |
| $ \int \frac {d x}{ x^2(ax + b)} = - \frac {1}{b x} + \frac {a}{b^2} \ln \left ( \frac {ax +b}{x} \right) $ | |
| $ \int \frac {d x}{ x^3(ax + b)} = \frac {2 a x - b}{2 b^2 x^2} + \frac {a^2}{b^3} \ln \left ( \frac {x}{ax+b} \right) $ | |
| $ \int \frac {d x}{(ax + b)^2} = \frac {-1}{a(ax+b)} $ | |
| $ \int \frac {x d x}{(ax + b)^2} = \frac {b}{a^2(ax+b)} + \frac {1}{a^2} \ln (ax+b) $ | |
| $ \int \frac {x^2 d x}{(ax + b)^2} = \frac {ax+b}{a^3} - \frac{b^2}{a^3(ax+b)} - \frac {2b}{a^3} \ln (ax+b) $ | |
| $ \int \frac {x^3 d x}{(ax + b)^2} = \frac {(ax+b)^2}{2a^4} - \frac {3b(ax+b)}{a^4} +\frac{b^3}{a^4(ax+b)} + \frac {3b^2}{a^4} \ln (ax+b) $ | |
| $ \int \frac {d x}{x(ax + b)^2} = \frac {1}{b(ax+b)} + \frac {1}{b^2} \ln \left ( \frac{x}{ax+b} \right ) $ | |
| $ \int \frac {d x}{x^2(ax + b)^2} = \frac {-a}{b^2(ax+b)} - \frac {1}{b^2x} + \frac {2a}{b^3} \ln \left ( \frac {ax+b}{x} \right ) $ | |
| $ \int \frac {d x}{x^3(ax + b)^2} = - \frac {(ax+b)^2}{2b^4x^2} + \frac {3 a(ax+b)}{b^4x} - \frac {a^3 x}{b^4(ax+b)} - \frac{3a^2}{b^4} \ln \left ( \frac {ax+b}{x} \right ) $ | |
| $ \int \frac {d x}{(ax + b)^3} = \frac {-1}{2(ax+b)^2} $ | |
| $ \int \frac {x d x}{(ax + b)^3} = \frac {-1}{a^2(ax+b)} + \frac {b}{2a^2(ax+b)^2} $ | |
| $ \int \frac {x^2 d x}{(ax + b)^3} = \frac {2b}{a^3(ax+b)} - \frac {b^2}{2a^3(ax+b)^2} + \frac {1}{a^3} \ln (ax+b) $ | |
| $ \int \frac {x^3 d x}{(ax + b)^3} = \frac {x}{a^3} - \frac {3b^2}{a^4(ax+b)} + \frac {b^3}{2a^4(ax+b)^2} - \frac {3b}{a^4} \ln (ax+b) $ | |
| $ \int \frac {d x}{x(ax + b)^3} = \frac {a^2x^2}{2b^3(ax+b)^2} - \frac {2ax}{b^3(ax+b)} - \frac {1}{b^3} \ln \left( \frac{ax+b}{x} \right) $ | |
| $ \int \frac {d x}{x^2(ax + b)^3} = \frac {-a}{2b^2(ax+b)^2} - \frac {2a}{b^3(ax+b)} - \frac {1}{b^3x} + \frac {3a}{b^4} \ln \left( \frac{ax+b}{x} \right) $ | |
| $ \int \frac {d x}{x^3(ax + b)^3} = \frac {a^4x^2}{2b^5(ax+b)^2} - \frac {4a^3x}{b^5(ax+b)} - \frac {(ax+b)^2}{2b^5x2} - \frac {6a^2}{b^5} \ln \left( \frac{ax+b}{x} \right) $ | |
| $ \int (a x +b)^n d x = \frac {(ax+b)^{n+1} }{(n+1)a}. \qquad n =-1 $ | |
| $ \int x (a x +b)^n d x = \frac {(ax+b)^{n+2} }{(n+2)a^2} - \frac {b(ax+b)^{n+1}}{(n+1)a^2}, \qquad n \neq -1,-2 $ | |
| $ \int x^2 (a x + b)^n d x = \frac {(ax+b)^{n+3} }{(n+3)a^3} - \frac {2b(ax+b)^{n+2}}{(n+2)a^3} + \frac {b^2(ax+b)^{n+1}}{(n+1)a^3} \qquad n = -1,-2, -3 $ | |
| $ \int x^m (a x + b)^n d x = \begin{cases} \frac {x^{m+1}(ax+b)^n}{m + n + 1} + \frac {n b}{m + n+ 1} \int x^m (ax+b)^{n-1} d x \\ \frac {x^m(ax+b)^{n+1}}{(m + n + 1)a} - \frac {m b}{(m + n+ 1)a} \int x^{m-1} (ax+b)^{n} d x \\ \frac {- x^{m+1}(ax+b)^{n+1}}{(n + 1)b} + \frac {m+ n+ 2 }{(n+ 1)b} \int x^m (ax+b)^{n+1} d x \end{cases} $ | |
| $ \text{ Particular Integral, component } \sqrt{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} $ | |
| $ \int \frac {d x}{x \sqrt {ax+b}} = \begin{cases} \frac {1}{b} \ln \left ( \frac {\sqrt {ax+b} - \sqrt {b}}{\sqrt {ax+b} + \sqrt {b}} \right ) \\ \frac {2}{\sqrt {-b}} \arctan \sqrt { \frac {ax+b}{- b}} \\ \end{cases} $ | |
| $ \int \frac { d x}{x ^2 \sqrt{a x + b}} = - \frac {\sqrt{ax+b}}{b x} - \frac {a}{2 b} \int \frac {d x}{x \sqrt {ax + b}} $ | |
| $ \int \sqrt{a x + b} \ d x = \frac {2 \sqrt{(ax+b)3}}{3 a} $ | |
| $ \int x \sqrt{a x + b} \ d x = \frac {2(3ax-2b)}{15a^2}\sqrt{(ax+b)^3} $ | |
| $ \int x^2 \sqrt{a x + b} \ d x = \frac {2(15a^2x^2-12abx + 8b^2)}{105a^3}\sqrt{(ax+b)^3} $ | |
| $ \int \frac {\sqrt {ax+b}}{x} \ d x = 2 \sqrt {ax+b} + b \ \int \frac {d x}{x \sqrt {ax + b}} $ | |
| $ \int \frac {\sqrt {ax+b}}{x^2} \ d x = - \frac {\sqrt {ax+b}}{x} + \frac {a}{2} \int \frac {d x}{x \sqrt {ax + b}} $ | |
| $ \int \frac {x^m}{\sqrt{ ax+b}} d x = \frac {2x^m \sqrt {ax+b}}{(2m+1)a} - \frac {2mb}{(2m+1)a} \int \frac {x^{m-1}}{\sqrt {ax+b}} d x $ | |
| $ \int \frac {d x}{x^m \sqrt{ax+b}} =- \frac {\sqrt{ax+b}}{(m-1)bx^{m-1}} - \frac {(2m-3)a}{(2m-2)b} \int \frac {d x}{x^{m-1} \sqrt{ax+b}} $ | |
| $ \int x^m \sqrt {ax+b} \ d x = \frac{2x^m}{(2m+3)a}(a+b)^{\frac{3}{2}} -\frac {2mb}{(2m+3)a} \int x^{m-1} \sqrt{ax+b} \ d x $ | |
| $ \int \frac {\sqrt {ax+b}}{x^m} d x = - \frac {\sqrt {ax+b}}{(m-1)x^{m-1}} + \frac {a}{2(m-1)} \int \frac {d x}{x^{m-1} \sqrt {ax+b}} $ | |
| $ \int \frac {\sqrt {ax+b}}{x^m} d x = \frac {-(ax+b)^{3/2}}{(m-1)bx^{m-1}} - \frac {(2m-5)a}{(2m-2)b} \int \frac {\sqrt {ax+b}}{x^{m-1}} d x $ | |
| $ \int (ax+b)^{m/2} d x = \frac {2(ax+b)^{(m+2)/2}}{a(m+2)} $ | |
| $ \int x(ax+b)^{m/2} d x = \frac {2(ax+b)^{(m+4)/2}}{a^2(m+4)} - \frac {2b(ax+b)^{(m+2)/2}}{a^2(m+2)} $ | |
| $ \int x^2(ax+b)^{m/2} d x = \frac {2(ax+b)^{(m+6)/2}}{a^3(m+6)} - \frac {4b(ax+b)^{(m+4)/2}}{a^3(m+4)}+ \frac {2b^2(ax+b)^{(m+2)/2}}{a^3(m+2)} $ | |
| $ \int \frac {(ax+b)^{m/2}}{x} d x =\frac {2(ax+b)^{m/2}}{m} + b \ \int \frac {(ax+b)^{(m-2)/2}}{x} d x $ | |
| $ \int \frac {(ax+b)^{m/2}}{x^2} d x = - \frac {(ax+b)^{(m+2)/2}}{bx} + \frac {ma}{2b} \ \int \frac {(ax+b)^{m/2}}{x} d x $ | |
| $ \int \frac {d x}{x(ax+b)^{m/2}} d x = \frac {2}{(m-2)b(ax+b)^{(m-2)/2}} + \frac {1}{b} \ \int \frac {d x}{x(ax+b)^{(m-2)/2}} $ | |
| Particular Integral, component ax + b and px + q | |
| $ \int \frac {d x}{(ax+b)(px+q)} = \frac {1}{bp-aq} \ln \left ( \frac {px+q}{ax+b} \right ) $ | |
| $ \int \frac {x d x}{(ax+b)(px+q)} = \frac {1}{bp-aq} \left \{ \frac{b}{a} \ln { (ax+b)} - \frac{q}{p} \ln{(px+q)} \right \} $ | |
| $ \int \frac { d x}{(ax+b)^2(px+q)} = \frac {1}{bp-aq} \left \{ \frac {1}{ax+b} + \frac {p}{bp- aq} \ln \left ( \frac {px+q}{ax +b} \right ) \right \} $ | |
| $ \int \frac {x d x}{(ax+b)^2(px+q)} = \frac {1}{bp-aq} \left \{ \frac {q}{bp- aq} \ln \left ( \frac {ax+b}{px +q} \right ) - \frac {b}{a(ax+b)} \right \} $ | |
| $ \int \frac {x^2 d x}{(ax+b)^2(px+q)} = \frac {b^2}{(bp -aq)a^2(ax+b)} + \frac {1}{(bp-aq)^2} \left \{ \frac {q^2}{p} \ln (px+q) + \frac {b(bp-2aq)}{a^2} \ln (ax+b) \right \} $ | |
| $ \int \frac { d x }{(ax+b)^m (px+q)^n } = \frac {-1}{(n-1)(bp-aq)} \left \{ \frac {1}{(ax+b)^{m-1}(pz+q)^{n-1} } + a(m + n -2) \ \int \frac {d x}{(ax+b)^m(px+q)^{n-1} } \right \} $ | |
| $ \int \frac {ax+b}{px+q} d x = \frac {ax}{p} + \frac {bp -aq}{p^2} \ln (px+q) $ | |
| $ \int \frac {(ax+b)^m}{(px+q)^n} d x = \begin{cases} \frac {-1}{(n-1)(bp-aq)} \left \{ \frac {(ax+b)^{m+1}}{(px+q)^{n-1}} + (n-m-2)a \int \frac {(ax+b)^m}{(px+q)^{n-1}} d x \right \} \\ \frac {-1}{(n-m-1)p} \left \{ \frac {(ax+b)^m}{(px+q)^{n-1}} + m(bp-aq) \int \frac{(ax+b)^{m-1}} {(px+q)^n} d x \right \} \\ \frac{-1}{(n-1)p} \left \{ \frac{(ax+b)^m}{(px+q)^{n-1}} - ma \int \frac{(ax+b)^{m-1}}{(px+q)^{n-1}} d x \right \} \end{cases} $ | |
| $ \text{Particular Integral, component } \sqrt{ax +b} \ and \ px+q $ | |
| $ \int \frac {px+q}{\sqrt {ax+b}} d x = \frac {2(apx+3aq-2bp)}{3a^2} \sqrt {ax+b} $ | |
| $ \int \frac {d x}{(px+q) \sqrt {ax+b}} = \begin{cases} \frac{1}{ \sqrt {bp-aq} \sqrt {p} } \ln \left ( \frac {\sqrt{p(ax+b)} -\sqrt {bp-aq} }{\sqrt{p(ax+b)} + \sqrt {bp-aq} } \right ) \\ \frac {2}{\sqrt {aq-bp} \sqrt {p}} \arctan \sqrt{ \frac {p(ax+b)}{aq-bp} } \end{cases} $ | |
| $ \int \frac {\sqrt {ax+b}}{px+q} \ d x = \begin{cases} \frac{2 \sqrt{ax+b}}{p} \ + \ \frac {\sqrt {bp-aq}}{p \sqrt{p}} \ln \left ( \frac {\sqrt {p(ax+b)} - \sqrt {bp-aq}}{\sqrt {p(ax+b)} + \sqrt {bp-aq}} \right ) \\ \frac {2 \sqrt {ax+b}}{p} \ - \ \frac {2 \sqrt{aq-bp}}{p \sqrt{p}} \arctan \sqrt { \frac {p(ax+b)}{aq-bp}} \\ \end{cases} $ | |
| $ \int (px+q)^n \sqrt {ax+b} \ d x = \frac{2(px+q)^{n+1} \sqrt {ax+b}}{(2n+3)p} \ + \ \frac {bp-aq}{(2n+3)p} \int \frac {(px+q)^n}{\sqrt {ax+b}} d x $ | |
| $ \int \frac {d x}{ (px+q)^n \sqrt {ax+b}} = \frac {\sqrt {ax+b}}{(n-1)(aq-bp)(px+q)^{n-1}} + \frac {(2n-3)a}{2(n-1)(aq-bp)} \int \frac {d x }{(px+q)^{n-1} \sqrt{ax+b}} $ | |
| $ \int \frac {(px+q)^n}{\sqrt {ax+b}} d x = \frac {2(px+q)^n \sqrt{ax+b}}{)2n+1)a} + \frac {2n(aq-bp)}{(2n+1)a} \int \frac {(px+q)^{n-1} d x}{\sqrt{ax+b}} $ | |
| $ \int \frac {\sqrt{ax+b}}{(px+q)^n} d x = \frac {- \sqrt {ax+b}}{(n-1)p(px+q)^n-1} + \frac{a}{2(n-1)p} \int \frac {d x}{(px+q)^{n-1} \sqrt {ax+b}} $ | |
| $ \text{Integeral Componant } \sqrt{ax+b} \text{ and } \sqrt{px+q} $ | |
| $ \int \frac {d x}{\sqrt {(ax+b)(px+q)}} = \begin{cases} \frac{2}{\sqrt{ap}} \ln { \left ( \sqrt {a(px+q)}+ \sqrt{p(ax+b)} \right )} \\ \frac {2}{\sqrt { -ap}} \arctan \ \sqrt { \frac {-p(ax+b)} {a(px+q)} } \\ \end{cases} $ | |
| $ \int \frac {x d x}{\sqrt {(ax+b)(px+q)}} = \frac {\sqrt{(ax+b)(px+q)}}{ap} - \frac{bp+aq}{2ap} \int \frac {d x}{\sqrt{(ax+b)(px+q)}} $ | |
| $ \int \sqrt {(ax+b)(px+q)} dx = \frac {2apx + bp +aq}{4ap} \sqrt {(ax+b)(px+q)} - \frac {(bp-aq)^2}{8ap} \int \frac{d x}{\sqrt{(ax+b)(px+q)}} $ | |
| $ \int \sqrt {\frac {px+q}{ax+b}} dx = \frac {\sqrt{(ax+b)(px+q)}}{a} + \frac {aq-bp}{2a} \int \frac {d x}{ \sqrt {(ax+b)(px+q)}} $ | |
| $ \int \frac {d x}{(px+q)\sqrt{(ax+b)(px+q)}} = \frac {2\sqrt{ax+b}}{(aq-bp)\sqrt{px+q}} $ | |
| $ \text {Integeral Componant } x^2 + a^2 $ | |
| $ \int \frac {d x}{ x^2 + a^2} \ = \ \frac {1}{a} \arctan \ \frac {x}{a} $ | |
| $ \int \frac {x \ d x}{x^2 + a^2} = \frac {1}{2} \ln \left ( x^2 + a^2 \right ) $ | |
| $ \int \frac {x^2 \ d x }{x^2 + a^2} = x \ - \ a \arctan \frac {x}{a} $ | |
| $ \int \frac {x^3 \ d x}{x^2 + a^2} = \frac{x^2}{2} - \frac{a^2}{2} \ln (x^2+a^2) $ | |
| $ \int \frac {d x}{ x(x^2 + a^2)} = \frac {1}{2a^2} \ln \left ( \frac {x^2}{x^2 + a^2} \right ) $ | |
| $ \int \frac {d x}{x^2(x^2 + a^2)} = - \frac {1}{a^2x} - \frac {1}{a^3} \arctan \frac {x}{a} $ | |
| $ \int \frac {d x}{x^3(x^2 + a^2)} = - \frac {1}{2a^2x^2} - \frac {1}{2a^4} \ln \left ( \frac {x^2}{x^2 + a^2} \right ) $ | |
| $ \int \frac {d x}{(x^2 +a^2)^2} =\frac {x}{2a^2(x^2+a^2)} + \frac {1}{2a^3} \arctan \frac {x}{a} $ | |
| $ \int \frac {x d x}{(x^2 +a^2)^2} = \frac {-1}{2(x^2 + a^2)} $ | |
| $ \int \frac {x^2 d x}{(x^2 +a^2)^2} = \frac {-x}{2(x^2 + a^2)} + \frac {1}{2a} \arctan \frac {x}{a} $ | |
| $ \int \frac {x^3 d x}{(x^2 +a^2)^2} = \frac {a^2}{2(x^2 + a^2)} + \frac{1}{2} \ln(x^2 + a^2) $ | |
| $ \int \frac {d x}{ x(x^2 + a^2)^2} = \frac {1}{2a^2(x^2+a^2)} + \frac{1}{2a^4} \ln \left ( \frac {x^2}{x^2 + a^2} \right ) $ | |
| $ \int \frac {d x}{x^2(x^2 + a^2)^2} = - \frac {1}{a^4x} - \frac {x}{2a^4(x^2 + a^2)} - \frac {3}{2a^5} \arctan \frac {x}{a} $ | |
| $ \int \frac {d x}{ x^3(x62 +a^2)^2} = - \frac {1}{2a^4x^2} - \frac {1}{2a^4(x^2+a^2)} - \frac {1}{a^6} \ln \left ( \frac {x^2}{x^2 + a^2} \right ) $ | |
| $ \int \frac {d x}{(x^2 + a^2)^n} = \frac {x}{2(n-1)a^2(x^2 + a^2)^{n-1}} + \frac {2n -3}{(2n - 2)a^2} \int \frac {d x}{(x^2 + a^2)^{n-1}} $ | |
| $ \int \frac {x dx}{(x^2+a^2)^n} = \frac {-1}{2(n-1)(x^2 + a^2)^{n-1}} $ | |
| $ \int \frac {d x}{x(x^2 +a^2)^n} = \frac {1}{2(n-1)a^2(x^2+a^2)^{n-1}} + \frac {1}{a^2} \int \frac {d x}{x(x^2 + a^2)^{n-1}} $ | |
| $ \int \frac {x^m d x}{(x^2 + a^2)^n} = \int \frac {x^{m-2} d x}{(x^2+a^2)^{n-1}} - a^2 \ \int \frac {x^{m-2} d x}{(x^2 + a^2)^n} $ | |
| $ \int \frac {d x}{x^m (x^2 +a^2)^n} = \frac {1}{a^2} \int \frac {d x}{x^m(x^2+a^2)^{n-1}} - \frac {1}{a^2} \int \frac {d x}{ x^{m-2}(x^2+a^2)^n} $ | |
| $ \text {Integeral Componant } x^2 - a^2 , x^2 > a^2 $ | |
| $ \int \frac {d x}{x^2 - a^2} = \frac {1}{2a} \ln \left ( \frac {x-a}{x+a} \right ) \text{ or } -\frac{1}{a} \operatorname{argcoth}\,\frac {x}{a} $ | |
| $ \int \frac {x d x}{x^2 - a^2} = \frac {1}{2} \ln(x^2 - a^2) $ | |
| $ \int \frac {x^2 d x}{x^2 -a^2} = x + \frac{a}{2} \ln \left ( \frac {x-a}{x+a} \right ) $ | |
| $ \int \frac {x^3 d x}{x^2 - a^2} = \frac{x^2}{2} + \frac {a^2}{2} \ln (x^2-a^2) $ | |
| $ \int \frac {d x}{x(x^2-a^2)} = \frac {1}{2a^2} \ln \left ( \frac {x^2 -a^2}{x^2} \right ) $ | |
| $ \int \frac {d x}{x^2(x^2-a^2)} = \frac {1}{a^2x} + \frac {1}{2a^3} \ln \left ( \frac {x-a}{x+a} \right ) $ | |
| $ \int \frac {d x}{x^3(x^2-a^2)} = \frac {1}{2a^2x^2} - \frac {1}{2a^4} \ln \left ( \frac {x^2}{x^2-a^2} \right ) $ | |
| $ \int \frac {d x}{(x^2 -a^2)^2} = \frac {-x}{2a^2(x^2-a^2)} - \frac {1}{4a^3} \ln \left ( \frac {x-a}{x+a} \right ) $ | |
| $ \int \frac {x d x}{(x^2 -a^2)^2} =\frac {-1}{2(x^2-a^2} $ | |
| $ \int \frac {x^2 d x}{(x^2 -a^2)^2} =\frac {-x}{2(x^2-a^2)} + \frac {1}{4a} \ln \left ( \frac {x-a}{x+a} \right ) $ | |
| $ \int \frac {x^3 d x}{(x^2 -a^2)^2} = \frac {-a^2}{2(x^2-a^2)} + \frac{1}{2} \ln (x^2-a^2) $ | |
| $ \int \frac {d x}{x(x^2 -a^2)^2} = \frac {-1}{2a^2(x^2-a^2)} + \frac {1}{2a^4} \ln \left ( \frac {x^2}{x^2-a^2} \right ) $ | |
| $ \int \frac {d x}{x^2(x^2 -a^2)^2} = - \frac {1}{a^4x} - \frac {x}{2a^4(x^2-a^2)} - \frac{3}{4a^5} \ln \left ( \frac {x-a}{x+a} \right ) $ | |
| $ \int \frac {d x}{x^3(x^2 -a^2)^2} = - \frac {1}{2a^4x^2} - \frac {1}{2a^4(x^2-a^2)} - \frac{1}{a^6} \ln \left ( \frac {x^2}{x^2-a^2} \right ) $ | |
| $ \int \frac {d x}{(x^2 -a^2)^n} = \frac {-x}{2(n-1)a^2(x^2-a^2)^{n-1}} - \frac {2n - 3}{(2n-2)a^2} \int \frac {d x}{(x^2-a^2)^{n-1}} $ | |
| $ \int \frac {x d x}{(x^2 -a^2)^n} = \frac {-1}{2(n-1)(x^2-a^2)^{n-1}} $ | |
| $ \int \frac {d x}{x(x^2 -a^2)^n} = \frac {-1}{2(n-1)a^2(x^2 - a^2)^{n-1}} - \frac {1}{a^2} \int \frac {d x}{x(x^2 -a^2)^{n-1}} $ | |
| $ \int \frac {x^m d x}{(x^2 -a^2)^n} = \int \frac {x^{m-2} d x}{(x^2 -a^2)^{n-1}} + a^2 \ \int \frac {x^{m-2} d x}{(x^2 -a^2)^n} $ | |
| $ \int \frac {d x}{x^m (x^2 -a^2)^n} =\frac {1}{a^2} \int \frac {d x}{x^{m-2} (x^2 -a^2)^n} - \frac{1}{a^2} \ \int \frac {d x}{x^m (x^2 -a^2)^{n-1}} $ | |
| 19 Integrals Component sin ax | |
| $ \int \sin a x d x = - \frac {\cos a x }{a} $ | |
| $ \int x \sin a x d x = \frac {\sin a x}{a^2}- \frac{x \cos a x}{a} $ | |
| $ \int x^2 \sin a x d x = \frac {2 x}{a^2} \sin a x + \left ( \frac {2}{a^3} - \frac {x^2}{a} \right)\cos a x $ | |
| $ \int x^3 \sin a x d x = \left( \frac {3 x^2}{a^2} - \frac{6}{a^4}\right)\sin a x + \left ( \frac {6x}{a^3} - \frac {x^3}{a} \right)\cos a x $ | |
| $ \int \frac {\sin a x}{x} d x = a x - \frac {(a x)^3}{3 \cdot 3!} + \frac {(a x)^5}{5 \cdot 5!} - \cdot \cdot \cdot $ | |
| $ \int \frac {\sin a x}{x^2} d x = - \frac {\sin a x}{x} + a \int \frac {\cos a x}{x}dx \qquad \left( \text {Voir} \text{ } 14.373 \right) $ | |
| $ \int \frac {d x}{\sin a x} = \frac {1}{a} \ln \left( \frac {1}{\sin a x} - \cot a x \right) = \frac {1}{a} \ln {\tan \frac{a x}{2}} $ | |
| $ \int \frac {x d x}{\sin a x} = \frac {1}{a^2} \left \{a x + \frac{(ax)^3}{18}+ \frac {7(ax)^5}{1800} + \cdot \cdot \cdot + \frac {2(2^{2n-1}-1)Bn(ax)^{2n+1}}{(2n+1)!} + \cdot\cdot\cdot \right \} $ | |
| $ \int \sin ^2 a x d x = \frac {x}{2}- \frac{\sin 2 a x}{4a} $ | |
| $ \int x \sin ^2 a x d x = \frac {x^2}{4}- \frac{x \sin 2 a x}{4a} - \frac {\cos 2 a x}{8a^2} $ | |
| $ \int \sin ^3 a x d x = -\frac {\cos a x}{a}- \frac{\cos^3 a x}{3a} $ | |
| $ \int \sin ^4 a x d x = \frac {3x}{8}- \frac{\sin 2 a x}{4a} + \frac {\sin4ax}{32a} $ | |
| $ \int \frac {d x}{\sin^2 a x} = -\frac {1}{a} \cot a x $ | |
| $ \int \frac {d x}{\sin^3 a x} = -\frac {\cos ax}{2a \sin^2 ax} + \frac{1}{2a}\ln \tan \frac{ax}{2} $ | |
| $ \int \sin px \sin q x d x = \frac {\sin (p-q)x}{2(p-q)} - \frac{\sin(p+q)x}{2(p+q)} \qquad \left( \text {Si} \text{ } p=\pm q, \text {voir}\text{ }14.368 \right) $ | |
| $ \int \frac {d x}{1-\sin a x} = \frac {1}{a} \tan {\left ( \frac{\pi}{4}+\frac{ax}{2}\right )} $ | |
| $ \int \frac {x d x}{1-\sin a x} = \frac {x}{a} \tan {\left ( \frac{\pi}{4}+\frac{ax}{2}\right )} +\frac{2}{a^2}\ln\sin{\left ( \frac{\pi}{4}-\frac{ax}{2}\right )} $ | |
| $ \int \frac {d x}{1+\sin a x} = -\frac {1}{a} \tan {\left ( \frac{\pi}{4}-\frac{ax}{2}\right )} $ | |
| $ \int \frac {x d x}{1+\sin a x} = -\frac {x}{a} \tan {\left ( \frac{\pi}{4}-\frac{ax}{2}\right )} +\frac{2}{a^2}\ln\sin{\left ( \frac{\pi}{4}+\frac{ax}{2}\right )} $ | |
| $ \int \frac {d x}{(1-\sin a x)^2} = \frac {1}{2a} \tan {\left ( \frac{\pi}{4}+\frac{ax}{2}\right )} +\frac{1}{6a}\tan^3{\left ( \frac{\pi}{4}+\frac{ax}{2}\right )} $ | |
| $ \int \frac {d x}{(1+\sin a x)^2} = -\frac {1}{2a} \tan {\left ( \frac{\pi}{4}-\frac{ax}{2}\right )} -\frac{1}{6a}\tan^3{\left ( \frac{\pi}{4}-\frac{ax}{2}\right )} $ | |
| $ \int \frac {dx}{(p+q \sin ax)}= \begin{cases} \frac{2}{a \sqrt {p^2-q^2}} \arctan \frac {p\tan \frac{1}{2}ax + q}{\sqrt{p^2-q^2}} \\ \frac{1}{a\sqrt{q^2-p^2}} \ln \left( \frac{p \tan \frac{1}{2} ax + q-\sqrt{q^2-p^2}}{p \tan \frac{1}{2}ax+q+\sqrt{q^2-p^2}} \right) \end{cases} \qquad \left ( \text{Si }p= \pm q \text{ voir }14.354 \text{ et } 14.356 \right ) $ | |
| $ \int \frac {d x}{(p+q \sin a x)^2}=\frac{q \cos ax}{a(p^2-q^2)(p+q \sin ax)}+\frac{p}{p^2-q^2}\int\frac{dx}{p+q \sin ax} \qquad \left ( \text{Si }p= \pm q \text{ voir }14.358 \text{ et } 14.359 \right ) $ | |
| $ \int \frac {d x}{p^2+q^2 \sin^2 a x}= \frac{1}{a p \sqrt {p^2+q^2}} \arctan \frac {\sqrt {p^2+q^2} \tan ax}{p} $ | |
| $ \int \frac {dx}{(p^2-q^2 \sin^2 a x)}= \begin{cases} \frac{1}{a p \sqrt {p^2-q^2}} \arctan \frac {\sqrt {p^2-q^2} \tan ax }{p} \\ \frac{1}{2 a p \sqrt {q^2-p^2}} \ln \left( \frac{ \sqrt{q^2-p^2} \tan ax + p}{\sqrt{q^2-p^2} \tan a x -p} \right) \end{cases} $ | |
| $ \int x^m \sin a x d x = -\frac {x^m \cos ax}{a} + \frac{m x^{m-1} \sin ax }{a^2} - \frac {m(m-1)}{a^2} \int x^{m-2} \sin ax dx $ | |
| $ \int \frac {\sin a x}{x^n} d x = - \frac {\sin a x}{(n-1) x^{n-1}} + \frac {a}{n-1} \int \frac {\cos ax }{x^{n-1}}dx \qquad \left ( \text{ Voir }14.395 \right ) $ | |
| $ \int \sin^n a x d x = -\frac {\sin^{n-1}ax \cos ax}{an} + \frac{n-1}{n} \int \sin^{n-2} ax dx $ | |
| $ \int \frac {dx}{\sin^n a x}= \frac { -\cos a x}{a(n-1) \sin^{n-1}ax} + \frac {n-2}{n-1} \int \frac {dx }{\sin^{n-2}ax} $ | |
| $ \int \frac {x dx}{\sin^n a x}= \frac {- x \cos a x}{a(n-1) \sin^{n-1}ax} - \frac {1}{a^2(n-2)(n-1)\sin^{n-2}ax} +\frac {n-2}{n-1} \int \frac {x dx }{\sin^{n-2}ax} $ | |
| 20 Integrals Component cos ax | |
| $ \int \cos a x d x = \frac {\sin a x }{a} $ | |
| $ \int x \cos a x d x = \frac {\cos a x}{a^2} + \frac{x \sin a x}{a} $ | |
| $ \int x^2 \cos a x d x = \frac {2 x}{a^2} \cos a x + \left ( \frac {x^2}{a} - \frac {2}{a^3} \right)\sin a x $ | |
| $ \int x^3 \cos a x d x = \left( \frac {3 x^2}{a^2} - \frac{6}{a^4}\right)\cos a x + \left ( \frac {x^3}{a} - \frac{6x}{a^3} \right)\sin a x $ | |
| $ \int \frac {\cos a x}{x} d x = \ln x - \frac {(a x)^2}{2 \cdot 2!} + \frac {(a x)^4}{4 \cdot 4!} - \frac {(a x)^6}{6 \cdot 6!} \cdot \cdot \cdot $ | |
| $ \int \frac {\cos a x}{x^2} d x = - \frac {\cos a x}{x} - a \int \frac {\sin a x}{x}dx \qquad \left( \text {Voir} \text{ } 14.343 \right) $ | |
| $ \int \frac {d x}{\cos a x} = \frac {1}{a} \ln \left( \frac {1}{\cos a x} +\tan a x \right)= \frac {1}{a} \tan {\left ( \frac{\pi}{4}+\frac{ax}{2}\right )} $ | |
| $ \int \frac {x d x}{\cos a x} = \frac {1}{a^2} \left \{\frac{(a x)^2}{2} + \frac{(ax)^4}{8}+ \frac {5(ax)^6}{144} + \cdot \cdot \cdot + \frac {En(ax)^{2n+2}}{(2n+2)(2n)!} + \cdot\cdot\cdot \right \} $ | |
| $ \int \cos ^2 a x d x = \frac {x}{2}+ \frac{\sin 2 a x}{4a} $ | |
| $ \int x \cos ^2 a x d x = \frac {x^2}{4}+ \frac{x \sin 2 a x}{4a} + \frac {\cos 2 a x}{8a^2} $ | |
| $ \int \cos ^3 a x d x = \frac {\sin a x}{a}- \frac{\sin^3 a x}{3a} $ | |
| $ \int \cos ^4 a x d x = \frac {3x}{8}+ \frac{\sin 2 a x}{4a} + \frac {\sin4ax}{32a} $ | |
| $ \int \frac {d x}{\cos^2 a x} = \frac {\cot a x}{a} $ | |
| $ \int \frac {d x}{\cos^3 a x} = \frac {\sin ax}{2a \cos^2 ax} + \frac{1}{2a}\ln \tan { \left( \frac{\pi}{4}+ \frac {ax}{2} \right)} $ | |
| $ \int \cos ax \cos p x d x = \frac {\sin (a-p)x}{2(a-p)} - \frac{\sin(a+p)x}{2(a+p)} \qquad \left( \text {Si } a=\pm p, \text { voir } 14.377 \right) $ | |
| $ \int \frac {d x}{1-\cos a x} = -\frac {1}{a} \cot \frac{ax}{2} $ | |
| $ \int \frac {x d x}{1-\cos a x} = -\frac {x}{a} \cot \frac{ax}{2} +\frac{2}{a^2}\ln\sin \frac{ax}{2} $ | |
| $ \int \frac {d x}{1+\cos a x} = \frac {1}{a} \tan \frac{ax}{2} $ | |
| $ \int \frac {x d x}{1+\cos a x} = \frac {x}{a} \tan \frac{ax}{2} + \frac{2}{a^2}\ln\cos \frac{ax}{2} $ | |
| $ \int \frac {d x}{(1-\cos a x)^2} = -\frac {1}{2a} \cot \frac{ax}{2} - \frac{1}{6a}\cot^3 \frac{ax}{2} $ | |
| $ \int \frac {d x}{(1+\cos a x)^2} = \frac {1}{2a} \tan \frac{ax}{2} + \frac{1}{6a}\tan^3 \frac{ax}{2} $ | |
| $ \int \frac {dx}{(p+q \cos ax)}= \begin{cases} \frac{2}{a \sqrt {p^2-q^2}} \arctan \sqrt {(p-q)/(p+q)} \tan \frac{1}{2}ax \\ \frac{1}{a \sqrt{q^2-p^2}} \ln \left( \frac{ \tan \frac{1}{2} ax + \sqrt{(q+p)/(q-p)}}{\tan \frac{1}{2}ax-\sqrt{(q+p)/(q-p)}} \right) \end{cases} \qquad \left ( \text{Si }p= \pm q \text{ voir }14.384 \text{ et } 14.386 \right ) $ | |
| $ \int \frac {d x}{(p+q \cos a x)^2}=\frac{q \sin ax}{a(q^2-p^2)(p+q \cos ax)}-\frac{p}{q^2-p^2}\int\frac{dx}{p+q \cos ax} \qquad \left ( \text{Si }p= \pm q \text{ voir }14.388 \text{ et } 14.389 \right ) $ | |
| $ \int \frac {d x}{p^2+q^2 \cos^2 a x}= \frac{1}{a p \sqrt {p^2+q^2}} \arctan \frac {p \tan ax}{\sqrt {p^2+q^2} } $ | |
| $ \int \frac {dx}{(p^2-q^2 \cos^2 a x)}= \begin{cases} \frac{1}{a p \sqrt {p^2-q^2}} \arctan \frac {p \tan ax }{\sqrt {p^2-q^2} } \\ \frac{1}{2 a p \sqrt {q^2-p^2}} \ln \left( \frac{ p \tan ax - \sqrt{q^2-p^2}}{p \tan a x + \sqrt{q^2-p^2}} \right) \end{cases} $ | |
| $ \int x^m \cos a x d x = -\frac {x^m \sin ax}{a} + \frac{m x^{m-1}}{a^2} \sin ax - \frac {m(m-1)}{a^2} \int x^{m-2} \cos ax dx $ | |
| $ \int \frac {\cos a x}{x^n} d x = - \frac {\cos a x}{(n-1) x^{n-1}} - \frac {a}{n-1} \int \frac {\sin ax }{x^{n-1}}dx \qquad \left ( \text{ Voir }14.365 \right ) $ | |
| $ \int \cos^n a x d x = -\frac {\sin ax \cos^{n-1}ax }{an} + \frac{n-1}{n} \int \cos^{n-2} ax dx $ | |
| $ \int \frac {dx}{\cos^n a x}= \frac { \sin a x \cos^{n-1}ax}{an} + \frac {n-2}{n-1} \int \frac {dx }{\cos^{n-2}ax} $ | |
| $ \int \frac {x dx}{\cos^n a x}= \frac {x \sin a x}{a(n-1) \cos^{n-1}ax} - \frac {1}{a^2(n-1)(n-2)\cos^{n-2}ax} +\frac {n-2}{n-1} \int \frac {x dx }{\cos^{n-2}ax} $ | |
| 21 Integrals Component sin ax ET cos ax | |
| $ \int \sin ax \cos a x d x = \frac {\sin^2 a x }{2a} $ | |
| $ \int \sin p x \cos q x d x = -\frac {\cos (p-q)x}{2(p-q)} - \frac{\cos(p+q)x}{2(p+q)} $ | |
| $ \int \sin^n x \cos a x d x = \frac{ \sin^{n+1} ax}{(n+1)a} \qquad \left ( \text{Si }n=-1, \text{ voir }14.440\right ) $ | |
| $ \int \cos^n x \sin a x d x = -\frac{ \cos^{n+1} ax}{(n+1)a} \qquad \left ( \text{Si }n=-1, \text{ voir }14.429 \right ) $ | |
| $ \int \sin^2 a x \cos^2 a x d x = \frac{x}{8} - \frac {\sin 4ax}{32a} $ | |
| $ \int \frac {d x}{\sin ax \cos a x} = \frac {1}{a} \tan ax $ | |
| $ \int \frac {d x}{\sin^2 ax \cos a x} = \frac {1}{a} \ln \tan { \left( \frac{\pi}{4}+ \frac {ax}{2} \right)} - \frac {1}{a \sin ax } $ | |
| $ \int \frac {d x}{\sin ax \cos^2 a x} = \frac {1}{a} \ln \tan \frac {ax}{2} + \frac {1}{a \cos ax } $ | |
| $ \int \frac {d x}{\sin^2 ax \cos^2 a x} = -\frac {2\cot 2 a x }{a} $ | |
| $ \int \frac {\sin^2 ax }{\cos a x}dx = -\frac {\sin a x}{a} +\frac {1}{a} \ln \tan { \left( \frac {ax}{2}+ \frac{\pi}{4} \right)} $ | |
| $ \int \frac {\cos^2 ax }{\sin a x}dx = \frac {\cos ax}{a} + \frac{1}{a} \ln \tan \frac{ax}{2} $ | |
| $ \int \frac {dx}{\cos ax (1 \pm \sin a x} = \mp \frac {1}{2a (1 \pm \sin ax} + \frac{1}{2a} \ln \tan { \left( \frac {ax}{2}+ \frac{\pi}{4} \right)} $ | |
| $ \int \frac {dx}{\sin ax (1 \pm \cos a x} = \pm \frac {1}{2a (1 \pm \cos ax} + \frac{1}{2a} \ln \tan \frac {ax}{2} $ | |
| $ \int \frac {dx}{\sin ax \pm \cos a x} = \frac {1}{a \sqrt {2}} \ln \tan { \left( \frac {ax}{2} \pm \frac{\pi}{8} \right)} $ | |
| $ \int \frac {\sin ax dx}{\sin ax \pm \cos a x} = \frac {x}{2} \mp \frac {1}{2a} \ln { \left( \sin ax \pm \cos ax \right)} $ | |
| $ \int \frac {\cos ax dx}{\sin ax \pm \cos a x} = \mp \frac {x}{2} + \frac {1}{2a} \ln { \left( \sin ax \pm \cos ax \right)} $ | |
| $ \int \frac {\sin ax dx}{p+q \cos a x} = -\frac {1}{aq} \ln { \left( p+ q\cos ax \right)} $ | |
| $ \int \frac {\cos ax dx}{p+q \cos a x} = \frac {1}{aq} \ln { \left( p+ q\sin ax \right)} $ | |
| $ \int \frac {\sin ax dx}{(p+q \cos a x)^n} = \frac {1}{aq(n-1)(p+q \cos ax)^{n-1}} $ | |
| $ \int \frac {\sin ax dx}{(p+q \sin a x)^n} = \frac {-1}{aq(n-1)(p+q \sin ax)^{n-1}} $ | |
| $ \int \frac {dx}{p \sin ax + q \cos a x} = \frac {1}{a \sqrt {p^2+q^2}} \ln \tan { \left( \frac {ax+\arctan (q/p)}{2} \right)} $ | |
| $ \int \frac {dx}{p \sin ax +q \cos ax + r}= \begin{cases} \frac{2}{a \sqrt {r^2 -p^2-q^2}} \arctan \left( \frac {p+(r-q) \tan (ax/2)}{\sqrt {r^2 - p^2-q^2}} \right) \\ \frac{1}{a \sqrt {q^2 +p^2-r^2}} \ln \left( \frac{ p- \sqrt{p^2+q^2-r^2}+(r+q)\tan \frac{ax}{2} }{p+\sqrt{p^2+q^2-r^2}+(r+q)\tan \frac{ax}{2}} \right) \end{cases} \left ( \text{ Si }r=q \text{ voir }14.421. \text{ Si } r^2=p^2+q^2 \text{ voir }14.422 \right ) $ | |
| $ \int \frac {dx}{p \sin ax +q (1+\cos ax )}= \frac{1}{ap} \ln \left( q+ p \tan \frac{ax}{2} \right) $ | |
| $ \int \frac {dx}{p \sin ax +q \cos ax \pm \sqrt {p^2+q^2}}= \frac {-1}{a \sqrt {p^2+q^2}} \tan { \left( \frac{\pi}{4} \mp \frac{ax+ \arctan{(q/p)}}{2} \right)} $ | |
| $ \int \frac {dx}{p^2 \sin^2 ax +q^2 \cos^2 ax }= \frac{1}{apq} \arctan { \frac {p \tan ax}{q}} $ | |
| $ \int \frac {dx}{p^2 \sin^2 ax - q^2 \cos^2 ax }= \frac{1}{2apq} \ln { \left( \frac {p \tan ax -q}{p \tan ax + q} \right)} $ | |
| $ \int \sin^m ax \cos^n ax dx= \begin{cases} -\frac{\sin^{m-1} ax \cos^{n-1}ax}{a (m+n)} + \frac {m-1}{m+n} \int \sin^{m-2} ax \cos^n ax dx \\ \frac{\sin^{m-1} ax \cos^{n+1}ax}{a (m+n)} + \frac{ n-1 }{m+n} \int \sin^m ax \cos^{n-2} ax dx \end{cases} $ | |
| $ \int \frac {\sin^m ax}{ \cos^n ax }dx= \begin{cases} \frac{\sin^{m-1} ax } {a(n-1) \cos^{n-1}ax} -\frac {m-1}{n-1} \int \sin^{m-2} ax \cos^{n-2} ax dx \\ \frac{\sin^{m+1} ax } {a(n-1) \cos^{n+1}ax} - \frac{m-n+2}{n-1} \int \sin^m ax \cos^{n-2} ax dx \\ \frac{- \sin^{m-1} ax } {a(m-n) \cos^{n-1}ax} + \frac {m-1}{m-n} \int \sin^{m-2} ax \cos^n ax dx \\ \end{cases} $ | |
| $ \int \frac {\cos^m ax}{ \sin^n ax }dx= \begin{cases} \frac{-\cos^{m-1} ax } {a(n-1) \sin^{n-1}ax} -\frac {m-1}{n-1} \int \cos^{m-2} ax \sin^{n-2} ax dx \\ \frac{-\cos^{m+1} ax } {a(n-1) \sin^{n-1}ax} - \frac{m-n+2}{n-1} \int \cos^m ax \sin^{n-2} ax dx \\ \frac{- \cos^{m-1} ax } {a(m-n) \sin^{n-1}ax} + \frac {m-1}{m-n} \int \cos^{m-2} ax \sin^n ax dx \\ \end{cases} $ | |
| $ \int \frac {dx}{ \sin^m ax \cos^n a x} \begin{cases} \frac{1}{a(n-1) \sin^{n-1}ax \cos^{n-1} ax} + \frac{m-n+2}{n-1} \frac {dx}{ \sin^m ax \cos^{n-2}} \\ \frac{-1}{a(n-1) \sin^{m-1}ax \cos^{n-1} ax} + \frac{m-n+2}{n-1} \frac {dx}{ \sin^{m-2} ax \cos^n ax}\\ \end{cases} $ | |
| 22 Integrals Component tan ax | |
| $ \int \tan a x d x = - \frac {1}{a} \ln {\cos a x } $ | |
| $ \int \tan ^2 a x d x = \frac { \tan ax}{a} - x $ | |
| $ \int \tan ^3 a x d x = \frac {\tan^2 ax}{2a}+ \frac{1}{a} \ln {\cos a x} $ | |
| $ \int \frac {\tan^n ax }{\cos^2 a x}dx = \frac {\tan^{n+1} a x}{(n+1)a} $ | |
| $ \int \frac {1}{\cos^2 a x \tan ax }dx = \frac {1}{a} \ln {\tan a x} $ | |
| $ \int \frac {dx}{ \tan ax } = \frac {1}{a} \ln {\sin a x} $ | |
| $ \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 \} $ | |
| $ \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 $ | |
| $ \int x \tan^2 ax dx = \frac {x \tan ax}{a} + \frac {1}{a^2} \ln {\cos a x} - \frac {x^2}{2} $ | |
| $ \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)} $ | |
| $ \int \tan^n ax dx = \frac {\tan^{n+1}ax}{(n+1)a} -\int \frac \tan^{n-2} a x dx $ | |
