Common Integrals

1.

$\displaystyle \int adx=ax$

2.

$\displaystyle \int af(x)dx=a \displaystyle \int f(x)dx$

3.

$\displaystyle \int \left( u \pm v \pm w \pm \cdots \right) dx = \displaystyle \int udx \pm \displaystyle \int vdx \pm \displaystyle \int wdx \pm \cdots$

4.

$\displaystyle \int udv = uv - \displaystyle \int vdu$

5.

$\displaystyle \int f(ax)dx = \displaystyle \frac{1}{a} \displaystyle \int f(u)du$

6.

$\displaystyle \int F\{f(x)\}dx = \displaystyle \int F(u) \displaystyle \frac{dx}{du}du = \displaystyle \int \displaystyle \frac{F(u)}{f'(x)}du$

7.

$\displaystyle \int u^{n}du = \displaystyle \frac{u^{n+1}}{n+1}, n \neq -1$

8.

$\begin{array}{lcl} \displaystyle \int\displaystyle \frac{du}{u} & = & \ln u \mb... ...or} \ln (-u) \mbox{ if} u<0 \\ & = & \ln \left\vert u \right\vert \end{array}$

9.

$\displaystyle \int e^{u}du=e^{u}$

10.

$\displaystyle \int a^{u}du = \int e^{u \ln a}du = \displaystyle \frac{e^{u \ln a}}{\ln a} = \displaystyle \frac{a^{u}}{\ln a} , a >0, a \neq 1$

11.

$\displaystyle \int \sin u du = -\cos u$

12.

$\displaystyle \int \cos u du = \sin u$

13.

$\displaystyle \int \tan u du = \ln \sec u = -\ln \cos u$

14.

$\displaystyle \int \cot u du = \ln \sin u$

15.

$\displaystyle \int \sec u du = \ln (\sec u + \tan u) = \ln \tan \left( \displaystyle \frac{u}{2} + \displaystyle \frac{\pi}{4} \right)$

16.

$\displaystyle \int \csc u du = \ln (\csc u - \cot u) = \ln \tan \displaystyle \frac{u}{2}$

17.

$\displaystyle \int \sec ^{2} u du = \tan u$

18.

$\displaystyle \int \csc ^{2} u du = -\cot u$

19.

$\displaystyle \int \tan ^{2} u du = \tan u - u$

20.

$\displaystyle \int \cot ^{2} u du = -\cot u - u$

21.

$\displaystyle \int \sin ^{2} u du = \displaystyle \frac{u}{2} - \displaystyle \frac{\sin 2u}{4} = \displaystyle \frac{1}{2} (u-\sin u \cos u)$

22.

$\displaystyle \int \cos ^{2} u du = \displaystyle \frac{u}{2} + \displaystyle \frac{\sin 2u}{4} = \displaystyle \frac{1}{2} (u+\sin u \cos u)$

23.

$\displaystyle \int \sec u \tan u du = \sec u$

24.

$\displaystyle \int \csc u \cot u du = -\csc u$

25.

$\displaystyle \int \sinh u du = \cosh u$

26.

$\displaystyle \int \cosh u du = \sinh u$

27.

$\displaystyle \int \tanh u du = \ln \cosh u$

28.

$\displaystyle \int \coth u du = \ln \sinh u$

29.

$\displaystyle \int$ sech $u du = \sin ^{-1}(\tanh u )$ or $2\tan ^{-1}e^{u}$

30.

$\displaystyle \int$ csch $u du = \ln \tanh \displaystyle \frac{u}{2}$ or $-\coth ^{-1}e^{u}$

31.

$\displaystyle \int$ sech $^{2} u du = \tanh u$

32.

$\displaystyle \int$ csch ² u du =-coth u

33.

$\displaystyle \int\tanh ^{2} u du = u - \tanh u$

34.

$\displaystyle \int$ coth ² u du = u -coth u

35.

$\displaystyle \int\sinh ^{2} u du = \displaystyle \frac{\sinh 2u}{4} - \displaystyle \frac{u}{2} = \displaystyle \frac{1}{2}(\sinh u \cosh u- u)$

36.

$\displaystyle \int\cosh ^{2} u du = \displaystyle \frac{\sinh 2u}{4} + \displaystyle \frac{u}{2} = \displaystyle \frac{1}{2}(\sinh u \cosh u+ u)$

37.

$\displaystyle \int$ sech $u \tanh u du = -$ sech u

38.

$\displaystyle \int$ csch ucoth u du = -csch u

39.

$\displaystyle \int\displaystyle \frac{du}{u^{2}+a^{2}} = \displaystyle \frac{1}{a} \tan^{-1} \displaystyle \frac{u}{a}$

40.

$\displaystyle \int\displaystyle \frac{du}{u^{2} - a^{2}}= \displaystyle \frac{1... ...n \left( \displaystyle \frac{u - a}{u+a} \right) = - \displaystyle \frac{1}{a}$ coth $^{-1} \displaystyle \frac{u}{a} , u^{2}>a^{2}$

41.

$\displaystyle \int\displaystyle \frac{du}{a^{2}-u^{2}}= \displaystyle \frac{1}{... ...= \displaystyle \frac{1}{a} \tanh ^{-1} \displaystyle \frac{u}{a} , u^{2}<a^{2}$

42.

$\displaystyle \int\displaystyle \frac{du}{\sqrt{a^{2}-u^{2}}} = \sin ^{-1} \displaystyle \frac{u}{a}$

43.

$\displaystyle \int\displaystyle \frac{du}{\sqrt{u^{2}+a^{2}}} = \ln \left( u+ \displaystyle\sqrt{u^{2} + a^{2}} \right)$ or $\sinh ^{-1} \displaystyle \frac{u}{a}$

44.

$\displaystyle \int\displaystyle \frac{du}{\sqrt{u^{2}-a^{2}}} = \ln \left( u + \displaystyle\sqrt{u^{2} - a^{2}} \right)$

45.

$\displaystyle \int\displaystyle \frac{du}{u \sqrt{u^{2}-a^{2}}} = \displaystyle \frac{1}{a} \sec ^{-1} \left\vert \displaystyle \frac{u}{a} \right\vert$

46.

$\displaystyle \int\displaystyle \frac{du}{u \sqrt{u^{2}+a^{2}}}=-\displaystyle \frac{1}{a} \ln \left( \displaystyle \frac{a+\sqrt{u^{2}+a^{2}}}{u} \right)$

47.

$\displaystyle \int\displaystyle \frac{du}{u \sqrt{a^{2}-u^{2}}}=-\displaystyle \frac{1}{a} \ln \left( \displaystyle \frac{a+\sqrt{a^{2}-u^{2}}}{u} \right)$

48.

$\displaystyle \int f^{(n)}gdx = f^{(n-1)}g - f^{(n-2)}g' + f^{(n-3)} g'' - \cdots (-1)^{n} \displaystyle \int fg^{(n)}dx$