Common Integrals

1.

$$\int adx=ax$$

2.

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

3.

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

4.

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

5.

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

6.

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

7.

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

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.

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

10.

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

11.

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

12.

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

13.

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

14.

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

15.

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

16.

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

17.

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

18.

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

19.

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

20.

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

21.

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

22.

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

23.

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

24.

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

25.

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

26.

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

27.

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

28.

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

29.

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

30.

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

31.

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

32.

$$\int$$csch ² u du =-coth u

33.

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

34.

$$\int$$coth ² u du = u -coth u

35.

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

36.

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

37.

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

38.

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

39.

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

40.

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

41.

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

42.

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

43.

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

44.

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

45.

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

46.

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

47.

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

48.

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

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