Indefinite Integral

${\displaystyle \int f(x)dx}$

Table of Properties of Integrals
	Rule	Conditions
1	${\displaystyle \int a\,dx=ax}$
2 Homogeniety	${\displaystyle \int af(x)\,dx=a\int f(x)\,dx}$
3 Associativity	${\displaystyle \int {\left(f\pm g\pm h\pm \cdots \right)\,dx}=\int f\,dx\pm \int g\,dx\pm \int h\,dx\pm \cdots }$
4 Integration by Parts	${\displaystyle \int _{a}^{b}fg'\,dx=\left[fg\right]_{a}^{b}-\int _{a}^{b}gf'\,dx}$
4 General Integration by Parts	${\displaystyle \int f^{(n)}g\,dx=f^{(n-1)}g'-f^{(n-2)}g''+\ldots +(-1)^{n}\int fg^{(n)}\,dx}$
5	${\displaystyle \int f(ax)\,dx={\frac {1}{a}}\int f(x)\,dx}$
6 Substitution Rule	${\displaystyle \int g\{f(x)\}\,dx=\int g(u){\frac {dx}{du}}\,du=\int {\frac {g(u)}{f'(x)}}\,du}$	${\displaystyle u=f(x)\,}$
7	${\displaystyle \int x^{n}\,dx={\frac {x^{n+1}}{n+1}}}$	${\displaystyle n\neq -1\,}$
8	${\displaystyle \int {\frac {1}{x}}\,dx=\ln |x|}$
9	${\displaystyle \int e^{x}\,dx=e^{x}}$
10	${\displaystyle \int a^{x}\,dx={\frac {a^{x}}{\ln |a|}}}$	${\displaystyle a\neq 1}$
Notes:	f, g, h are functions of x a, n are constants. The constant of integration, C has been omitted from this table. It should be included in the working of the equation if applicable.

	Integral	Value	Remarks
1	${\displaystyle \int c\,dx}$	${\displaystyle cx+C\,}$
2	${\displaystyle \int x^{n}\,dx}$	${\displaystyle {\frac {x^{n+1}}{n+1}}+C}$	${\displaystyle n\neq -1}$
3	${\displaystyle \int {\frac {1}{x}}\,dx}$	${\displaystyle \ln {\left|x\right|}+C}$
4	${\displaystyle \int {1 \over {a^{2}+x^{2}}}\,dx}$	${\displaystyle {1 \over a}\arctan {x \over a}+C}$
5	${\displaystyle \int {1 \over {\sqrt {a^{2}-x^{2}}}}\,dx}$	${\displaystyle \arcsin {x \over a}+C}$
6	${\displaystyle \int {-1 \over {\sqrt {a^{2}-x^{2}}}}\,dx}$	${\displaystyle \arccos {x \over a}+C}$
7	${\displaystyle \int {1 \over x{\sqrt {x^{2}-a^{2}}}}\,dx}$	${\displaystyle {1 \over a}{\mbox{arcsec}}\,{|x| \over a}+C}$
8	${\displaystyle \int \ln {x}\,dx}$	${\displaystyle x\ln {x}-x+C\,}$
9	${\displaystyle \int \log _{b}{x}\,dx}$	${\displaystyle x\log _{b}{x}-x\log _{b}{e}+C\,}$
10	${\displaystyle \int e^{x}\,dx}$	${\displaystyle e^{x}+C\,}$
11	${\displaystyle \int a^{x}\,dx}$	${\displaystyle {\frac {a^{x}}{\ln {a}}}+C}$
12	${\displaystyle \int \sin {x}\,dx}$	${\displaystyle -\cos {x}+C\,}$
13	${\displaystyle \int \cos {x}\,dx}$	${\displaystyle \sin {x}+C\,}$
14	${\displaystyle \int \tan {x}\,dx}$	${\displaystyle -\ln {\left|\cos {x}\right|}+C\,}$
15	${\displaystyle \int \cot {x}\,dx}$	${\displaystyle \ln {\left|\sin {x}\right|}+C\,}$
16	${\displaystyle \int \sec {x}\,dx}$	${\displaystyle \ln {\left|\sec {x}+\tan {x}\right|}+C\,}$
17	${\displaystyle \int \csc {x}\,dx}$	${\displaystyle -\ln {\left|\csc {x}+\cot {x}\right|}+C\,}$
18	${\displaystyle \int \sec ^{2}x\,dx}$	${\displaystyle \tan x+C\,}$
19	${\displaystyle \int \csc ^{2}x\,dx}$	${\displaystyle -\cot x+C\,}$
20	${\displaystyle \int \sec {x}\,\tan {x}\,dx}$	${\displaystyle \sec {x}+C\,}$
21	${\displaystyle \int \csc {x}\,\cot {x}\,dx}$	${\displaystyle -\csc {x}+C\,}$
22	${\displaystyle \int \sin ^{2}x\,dx}$	${\displaystyle {\frac {1}{2}}(x-\sin x\cos x)+C\,}$
23	${\displaystyle \int \cos ^{2}x\,dx}$	${\displaystyle {\frac {1}{2}}(x+\sin x\cos x)+C\,}$
24	${\displaystyle \int \sin ^{n}x\,dx}$	${\displaystyle -{\frac {\sin ^{n-1}{x}\cos {x}}{n}}+{\frac {n-1}{n}}\int \sin ^{n-2}{x}\,dx}$
25	${\displaystyle \int \cos ^{n}x\,dx}$	${\displaystyle -{\frac {\cos ^{n-1}{x}\sin {x}}{n}}+{\frac {n-1}{n}}\int \cos ^{n-2}{x}\,dx}$
26	${\displaystyle \int \arctan {x}\,dx}$	${\displaystyle x\,\arctan {x}-{\frac {1}{2}}\ln {\left|1+x^{2}\right|}+C}$
27	${\displaystyle \int \sinh x\,dx}$	${\displaystyle \cosh x+C\,}$
28	${\displaystyle \int \cosh x\,dx}$	${\displaystyle \sinh x+C\,}$
29	${\displaystyle \int \tanh x\,dx}$	${\displaystyle \ln |\cosh x|+C\,}$
30	${\displaystyle \int {\mbox{csch}}\,x\,dx}$	${\displaystyle \ln \left|\tanh {x \over 2}\right|+C}$
31	${\displaystyle \int {\mbox{sech}}\,x\,dx}$	${\displaystyle \arctan(\sinh x)+C\,}$
32	${\displaystyle \int \coth x\,dx}$	${\displaystyle \ln |\sinh x|+C\,}$

Definite Integral

${\displaystyle \int _{a}^{b}f(x),dx}$

${\displaystyle \int _{a}^{b}f(x)\,dx=F(b)-F(a).}$

Furthermore, for every x in the interval (a, b),

${\displaystyle {\frac {d}{dx}}\int _{a}^{x}f(t)\,dt=f(x).}$