Table of Integrals

This is a short table of some standard integrals. The left column is the function and the right column is an indefinite integral for it. You can add in a constant of integration if you want to.

xⁿ	$\displaystyle {\frac{1}{n+1}}$ x^{n + 1}	n $\ne$ - 1
$\displaystyle {\frac{1}{x}}$	ln\|x\|	see text for problems
$\displaystyle {\frac{1}{ax+b}}$	$\displaystyle {\frac{1}{a}}$ ln\|ax + b\|	as above
$\displaystyle {\frac{x}{ax+b}}$	$\displaystyle {\frac{1}{a^2}}$ $\displaystyle \left(\vphantom{ ax+b-b\ln\vert ax+b\vert}\right.$ ax + b - b ln\|ax + b\| $\displaystyle \left.\vphantom{ ax+b-b\ln\vert ax+b\vert}\right)$	as above
$\displaystyle {\frac{1}{a^2 + x^2}}$	$\displaystyle {\frac{1}{a}}$ arctan(x/a)
$\displaystyle {\frac{1}{\sqrt{a^2-x^2}}}$	arcsin(x/a)	0 $\le$ x $\le$ a
$\displaystyle {\frac{1}{\sqrt{x^2-a^2}}}$	ln(x + $\displaystyle \sqrt{x^2-a^2}$ )	0 $\le$ a $\le$ x
$\displaystyle {\frac{1}{\sqrt{x^2+a^2}}}$	ln(x + $\displaystyle \sqrt{x^2+a^2}$ )
sin x	-cos x
cos x	sin x
sec²x	tan x
$\displaystyle \cosec^{2}_{}$ x	-cot x
tan x	-lncos x
sec x	ln(sec x + tan x)
$\displaystyle \cosec$ x	ln( $\displaystyle \cosec$ x - cot x)
sin²x	$\displaystyle {\textstyle\frac{1}{2}}$ x - $\displaystyle {\textstyle\frac{1}{4}}$ sin(2x)
cos²x	$\displaystyle {\textstyle\frac{1}{2}}$ x + $\displaystyle {\textstyle\frac{1}{4}}$ sin(2x)
tan²x	tan x - x

sin(ax)sin(bx)	$\displaystyle {\textstyle\frac{1}{2}}$ $\displaystyle \left(\vphantom{\frac{\sin(a-b)x}{a-b} - \frac{\sin(a+b)x}{a+b} }\right.$ $\displaystyle {\frac{\sin(a-b)x}{a-b}}$ - $\displaystyle {\frac{\sin(a+b)x}{a+b}}$ $\displaystyle \left.\vphantom{\frac{\sin(a-b)x}{a-b} - \frac{\sin(a+b)x}{a+b} }\right)$	a $\ne$ ±b
sin(ax)cos(bx)	- $\displaystyle {\textstyle\frac{1}{2}}$ $\displaystyle \left(\vphantom{\frac{\cos(a-b)x}{a-b} + \frac{\cos(a+b)x}{a+b} }\right.$ $\displaystyle {\frac{\cos(a-b)x}{a-b}}$ + $\displaystyle {\frac{\cos(a+b)x}{a+b}}$ $\displaystyle \left.\vphantom{\frac{\cos(a-b)x}{a-b} + \frac{\cos(a+b)x}{a+b} }\right)$	a $\ne$ ±b
cos(ax)cos(bx)	$\displaystyle {\textstyle\frac{1}{2}}$ $\displaystyle \left(\vphantom{\frac{\sin(a-b)x}{a-b} + \frac{\sin(a+b)x}{a+b} }\right.$ $\displaystyle {\frac{\sin(a-b)x}{a-b}}$ + $\displaystyle {\frac{\sin(a+b)x}{a+b}}$ $\displaystyle \left.\vphantom{\frac{\sin(a-b)x}{a-b} + \frac{\sin(a+b)x}{a+b} }\right)$	a $\ne$ ±b
x sin(ax)	$\displaystyle {\frac{1}{a^2}}$ (sin(ax) - ax cos(ax))
x cos(ax)	$\displaystyle {\frac{1}{a^2}}$ (cos(ax) + ax sin(ax))
arcsin x	x arcsin x + $\displaystyle \sqrt{1-x^2}$
arccos x	x arccos x - $\displaystyle \sqrt{1-x^2}$
arctan x	x arctan x - $\displaystyle {\textstyle\frac{1}{2}}$ ln(1 + x²)
e^{ax + b}	$\displaystyle {\frac{1}{a}}$ e^{ax + b}
xe^ax	$\displaystyle {\frac{ax-1}{a^2}}$ e^ax
e^axsin(bx)	$\displaystyle {\frac{e^{ax}}{a^2 +b^2}}$ (a sin(bx) - b cos(bx))
e^axcos(bx)	$\displaystyle {\frac{e^{ax}}{a^2 +b^2}}$ (a cos(bx) + b sin(bx))
sinh x	cosh x
cosh x	sinh x
tanh x	lncosh x
ln x	x ln x - x
x ln(x)	x²( $\displaystyle {\textstyle\frac{1}{2}}$ ln x - $\displaystyle {\textstyle\frac{1}{4}}$ )
$\displaystyle {\frac{\ln x}{x}}$	$\displaystyle {\textstyle\frac{1}{2}}$ ln²x
$\displaystyle {\frac{1}{x\ln x}}$	ln(ln(x))