Indefinite Integrals

1. Basic rules of integration

1) If $F'(x)=f(x)$ , then

$\int f(x) dx=F(x)+C$

Where $C$ is an arbitrary constant.

2) $\int{Af(x)dx}=A\int{f(x)dx}$ , where $A$ is a constant quantity.

3) $\int \left[ f_1(x) \pm f_2(x) dx \right] =\int f_1(x) dx\pm \int f_2(x) dx$

4) If $\int f(x) dx=F(x)+C$ and $u=\phi (x)$ , then

$\int f(u) dx=F(u)+C$

In particular,

$\displaystyle \int f(ax+b) dx=\frac{1}{a}F(ax+b)+C$

2. Table of standard integrals

I.	$\displaystyle \int x^n dx = \frac{x^{n+1}}{n+1}; n\ne -1$
II.	$\displaystyle \int \frac{dx}{x} =\ln \|x\|+C$
III.	$\displaystyle \int \frac{dx}{x^2+a^2} dx =\frac{1}{a}\arctan \frac{x}{a}+C; (a \ne 0)$
IV.	$\displaystyle \int \frac{dx}{x^2-a^2} =\frac{1}{2a} \ln \left\|\frac{x-a}{x+a}\right\|+C; (a \ne 0)$
	$\displaystyle \int \frac{dx}{a^2-x^2} =\frac{1}{2a} \ln \left\|\frac{a+x}{a-x}\right\|+C ; (a \ne 0)$
V.	$\displaystyle \int \frac{dx}{\sqrt{x^2+a}} = \ln \left\|x+\sqrt{x^2+a}\right\|+C ; (a \ne 0)$
VI.	$\displaystyle \int \frac{dx}{\sqrt{a^2-x^2}} = \arcsin \frac{x}{a}+C=-\arccos \frac{x}{a}+C; a>0$
VII.	$\displaystyle \int a^xdx = \frac{a^x}{\ln a}+C, a>0; \int e^xdx=e^x+C$
VIII.	$\int \sin x dx =-\cos x +C$
IX.	$\int \cos x dx =\sin x +C$
X.	$\displaystyle \int \frac{dx}{\cos^2 x}=\tan x +C$
XI.	$\displaystyle \int \frac{dx}{\sin^2 x}=-\cot x +C$
XII.	$\displaystyle \int \frac{dx}{\sin x}=\ln \left\| \tan \frac{x}{2}\right\|+C=\ln \left\| \frac{1}{\sin x}-\cot x \right\|+C$
XIII	$\displaystyle \int \frac{dx}{\cos x}=\ln \left\| \tan \left(\frac{x}{2}+\frac{\pi}{4}\right)\right\|+C=\ln \left\| \frac{1}{\cos x}+\tan x \right\|+C$
XIV.	$\displaystyle \int \sinh x dx =\cosh x +C; \sinh x =\frac{1}{2}\left(e^x-e^{-x}\right); \cosh x =\frac{1}{2}\left(e^x+e^{-x}\right)$
XV.	$\int \cosh x dx =\sinh x +C$
XVI.	$\displaystyle \int \frac{dx}{\cosh^2 x}=\tanh x +C; \tanh x =\frac{\sinh x}{\cosh x}$
XVII.	$\displaystyle \int \frac{dx}{\sinh^2 x}=-\coth x +C; \coth x =\frac{\cosh x}{\sinh x}$