Integration

If $\frac{d}{dx}$ [ f (x)] = $f^{'}$ (x), then integral of $f^{'}$ (x), denoted by $\int f^{'}$ (x) dx, is given by f(x)
+c, where 'c' is a constant called the constant of integration.

1.	$\int dx = x + c$	2.	$\int x^{n} dx = \frac{x^{n + 1}}{n + 1} + c, n \neq - 1$
3.	$\int (1 / x) dx = \log \| x \| + c$	4.	$\int$ $e^{x} dx = e^{x}$ + c
5.	$\int$ $a^{x} dx = \frac{a^{x}}{\log_{e} a} + c$	6.	$\int \frac{1}{x^{2}} dx = \frac{- 1}{x}$ +c
7.	$\int \frac{1}{2 \sqrt{x}}$ dx= $\sqrt{x} + c$	8.	$\int k d x = kx + c$
9.	$\int \sin x dx = - \cos x + c$	10.	$\int \cos x d x = \sin x + c$

11. $\int$ tan x dx = log $| \sec x |$ + c or -log $| \cos x |$ + c

12. $\int$ sec x dx = log $| \sec x + \tan x |$ + c or log $| \tan (\frac{π}{4} + \frac{x}{2}) |$ + c

13. $\int$ cos ecx dx = log |cos ecx - cot x| + c or log $| \tan (x / 2) + c |$

14. $\int$ cot x dx = log | sin x| + c
15. $\int$ $\sec^{2}$ xdc = tan x + c
16. $\int$ cos ${ec}^{2}$ xdx= -cot x +c
17. $\int$ sec x. tan xdx= sec x + c
18. $\int$ cos ecx. cot xdx = -cos ecx + c
19. $\int \frac{1}{\sqrt{1 + x^{2}}}$ dx= $\sin^{- 1} x + c$ or - $\cos^{- 1} x + c$
20. $\int \frac{1}{1 + x^{2}}$ dx= $\tan^{- 1}$ x+c or - $\cot^{- 1} x + c$
21. $\int$ $\frac{1}{x \sqrt{x^{2} - 1}}$ dx= $\sec^{- 1}$ x + c or - ${cosec}^{- 1}$ x + c
22. $\int \frac{1}{a^{2} + x^{2}} dx = \frac{1}{a}$ $\tan^{- 1}$ (x/a) +c

23. $\int \frac{1}{a^{2} - x^{2}} dx = \frac{1}{2 a} \log | \frac{a + x}{a - x} |$ +c

24. $\int \frac{1}{x^{2} - a^{2}}$ dx $\frac{1}{2 a}$ log $| \frac{x - a}{x + a} |$ +c

25. $\int \frac{1}{\sqrt{a^{2} + x^{2}}} dx = \log | x + \sqrt{x^{2} + a^{2}} | + c or \sin h^{- 1}$ (x/a) +c

26. $\int \frac{1}{\sqrt{a^{2} - x^{2}}} dx = \sin^{- 1} (x / a) + c$

27. $\int \frac{1}{\sqrt{x^{2} - a^{2}}} dx = \log | x + \sqrt{x^{2} - a^{2}} |$ +c or cos $h^{- 1} (x / a) + c$

28. $\int \sqrt{a^{2} + x^{2}} dx = \frac{x}{2}$ $\sqrt{a^{2} + x^{2}}$ + $\frac{a^{2}}{2}$ log $| x + \sqrt{a^{2} + x^{2}} | + c$

29. $\int \sqrt{a^{2} + x^{2}} dx = \frac{x}{2}$ $\sqrt{a^{2} - x^{2}}$ + $\frac{a^{2}}{2}$ $\sin^{- 1}$ (x/a) +c

30. $\int \sqrt{x^{2} - a^{2}}$ dx= $\frac{x}{2}$ $\sqrt{x^{2} - a^{2}} - \frac{a^{2}}{2}$ log $| x + \sqrt{x^{2} - a^{2}} | + c$
31. $\int \frac{| x |}{x}$ dx = |x|

Rule to integrate f(ax+b) where a, b, are constants:

If x be replaced by (ax+b) on both sides of any standard result, the standard form remains true, provided the result on R.H.S is divided by a, the coefficient of x.

$\int$ sin (ax +b) dx = $\frac{\cos (ax + b)}{a}$ +c;

$\int {(ax + b)}^{n}$ dx= $\frac{{(ax + b)}^{n + 1}}{a (n + 1)}$ +c

$\int \frac{f^{'} (x)}{f (x)}$ = log |f(x)| + c and $\int \frac{f^{'} (x)}{2 \sqrt{f (x)}}$ dx= $\sqrt{f (x)}$ +c

$\int {[f (x)]}^{n}$ . $f^{'} (x) dx = \frac{{[(f (x))]}^{n + 1}}{n + 1} + c, n \neq - 1$

Rule to Integrate by parts

If u and v are any two functions of x, then $\int$ uv dx = u $\int$ $(\frac{du}{dx} . \int vdx)$ dx

ie, Integral of the product of two functions = 1st function x Integral of 2nd function -

Integral of (Derivative of 1st x Integral of 2nd)

Rule to evaluate

$\int$ $\frac{dx}{a + b \sin x}$ or $\int \frac{dx}{a + b \cos x}$ or $\int$ $\frac{dx}{a + b \cos x + c \sin x}$

Put tan (x/2) = u,

dx = $\frac{2 du}{1 + u^{2}}$ , sin x= $\frac{2 u}{1 + u^{2}}$ , cos x= $\frac{1 - u^{2}}{1 + u^{2}}$

Substitute these values of sin x ( or cos x) and dx in the given integral and then

integrate.