Engineering Handbook/Mathematics/Integral

Integral

Indefinite Integral

$\int f(x)dx$

Table of Properties of Integrals
	Rule	Conditions
1	$\int a\,dx=ax$
2 Homogeniety	$\int af(x)\,dx=a\int f(x)\,dx$
3 Associativity	$\int {\left(f\pm g\pm h\pm \cdots \right)\,dx}=\int f\,dx\pm \int g\,dx\pm \int h\,dx\pm \cdots$
4 Integration by Parts	$\int _{a}^{b}fg'\,dx=\left[fg\right]_{a}^{b}-\int _{a}^{b}gf'\,dx$
4 General Integration by Parts	$\int f^{(n)}g\,dx=f^{(n-1)}g'-f^{(n-2)}g''+\ldots +(-1)^{n}\int fg^{(n)}\,dx$
5	$\int f(ax)\,dx={\frac {1}{a}}\int f(x)\,dx$
6 Substitution Rule	$\int g\{f(x)\}\,dx=\int g(u){\frac {dx}{du}}\,du=\int {\frac {g(u)}{f'(x)}}\,du$	$u=f(x)\,$
7	$\int x^{n}\,dx={\frac {x^{n+1}}{n+1}}$	$n\neq -1\,$
8	$\int {\frac {1}{x}}\,dx=\ln |x|$
9	$\int e^{x}\,dx=e^{x}$
10	$\int a^{x}\,dx={\frac {a^{x}}{\ln |a|}}$	$a\neq 1$
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	$\int c\,dx$	$cx+C\,$
2	$\int x^{n}\,dx$	${\frac {x^{n+1}}{n+1}}+C$	$n\neq -1$
3	$\int {\frac {1}{x}}\,dx$	$\ln {\left|x\right|}+C$
4	$\int {1 \over {a^{2}+x^{2}}}\,dx$	${1 \over a}\arctan {x \over a}+C$
5	$\int {1 \over {\sqrt {a^{2}-x^{2}}}}\,dx$	$\arcsin {x \over a}+C$
6	$\int {-1 \over {\sqrt {a^{2}-x^{2}}}}\,dx$	$\arccos {x \over a}+C$
7	$\int {1 \over x{\sqrt {x^{2}-a^{2}}}}\,dx$	${1 \over a}{\mbox{arcsec}}\,{|x| \over a}+C$
8	$\int \ln {x}\,dx$	$x\ln {x}-x+C\,$
9	$\int \log _{b}{x}\,dx$	$x\log _{b}{x}-x\log _{b}{e}+C\,$
10	$\int e^{x}\,dx$	$e^{x}+C\,$
11	$\int a^{x}\,dx$	${\frac {a^{x}}{\ln {a}}}+C$
12	$\int \sin {x}\,dx$	$-\cos {x}+C\,$
13	$\int \cos {x}\,dx$	$\sin {x}+C\,$
14	$\int \tan {x}\,dx$	$-\ln {\left|\cos {x}\right|}+C\,$
15	$\int \cot {x}\,dx$	$\ln {\left|\sin {x}\right|}+C\,$
16	$\int \sec {x}\,dx$	$\ln {\left|\sec {x}+\tan {x}\right|}+C\,$
17	$\int \csc {x}\,dx$	$-\ln {\left|\csc {x}+\cot {x}\right|}+C\,$
18	$\int \sec ^{2}x\,dx$	$\tan x+C\,$
19	$\int \csc ^{2}x\,dx$	$-\cot x+C\,$
20	$\int \sec {x}\,\tan {x}\,dx$	$\sec {x}+C\,$
21	$\int \csc {x}\,\cot {x}\,dx$	$-\csc {x}+C\,$
22	$\int \sin ^{2}x\,dx$	${\frac {1}{2}}(x-\sin x\cos x)+C\,$
23	$\int \cos ^{2}x\,dx$	${\frac {1}{2}}(x+\sin x\cos x)+C\,$
24	$\int \sin ^{n}x\,dx$	$-{\frac {\sin ^{n-1}{x}\cos {x}}{n}}+{\frac {n-1}{n}}\int \sin ^{n-2}{x}\,dx$
25	$\int \cos ^{n}x\,dx$	$-{\frac {\cos ^{n-1}{x}\sin {x}}{n}}+{\frac {n-1}{n}}\int \cos ^{n-2}{x}\,dx$
26	$\int \arctan {x}\,dx$	$x\,\arctan {x}-{\frac {1}{2}}\ln {\left|1+x^{2}\right|}+C$
27	$\int \sinh x\,dx$	$\cosh x+C\,$
28	$\int \cosh x\,dx$	$\sinh x+C\,$
29	$\int \tanh x\,dx$	$\ln |\cosh x|+C\,$
30	$\int {\mbox{csch}}\,x\,dx$	$\ln \left|\tanh {x \over 2}\right|+C$
31	$\int {\mbox{sech}}\,x\,dx$	$\arctan(\sinh x)+C\,$
32	$\int \coth x\,dx$	$\ln |\sinh x|+C\,$

Definite Integral

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

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

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

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

References

1. Integral