| 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 2 arc th e^u $ | |
| $ \int \frac {1}{\operatorname{sh}\ u} \ d u = \ln \operatorname{th}\,\frac{2}{2} \qquad - 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 \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} $ | |
| 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 ET 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 } $ | |
| 19 Integrals Including, 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} $ | |
