Engineering Handbook/Mathematics/Integral

Integral

Indefinite Integral

$\int f(x)dx$

Table of Properties of Integrals
	Rule	Conditions
1	$\int adx=ax$
2 Homogeniety	$\int af(x)dx=a\int f(x)dx$
3 Associativity	$\int (f\pm g\pm h\pm \cdots )dx=\int fdx\pm \int gdx\pm \int hdx\pm \cdots$
4 Integration by Parts	$\underset{a}{\overset{b}{\int }}f{g}^{\prime }dx={\left[fg\right]}_a^b-\underset{a}{\overset{b}{\int }}g{f}^{\prime }dx$
4 General Integration by Parts	$\int f^{(n)}gdx=f^{(n-1)}{g}^{\prime }-f^{(n-2)}{g}^{″}+\dots +(-1{)}^n\int f{g}^{(n)}dx$
5	$\int f(ax)dx=\frac{1}{a}\int f(x)dx$
6 Substitution Rule	$\int g\{f(x)\}dx=\int g(u)\frac{dx}{du}du=\int \frac{g(u)}{f^{\prime }(x)}du$	$u=f(x)$
7	$\int x^{n}dx=\frac{x^{n+1}}{n+1}$	$n\ne -1$
8	$\int \frac{1}{x}dx=\ln |x|$
9	$\int e^{x}dx=e^x$
10	$\int a^{x}dx=\frac{a^x}{\ln |a|}$	$a\ne 1$
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.
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	Integral	Value	Remarks
1	$\int cdx$	$cx+C$
2	$\int x^{n}dx$	$\frac{x^{n+1}}{n+1}+C$	$n\ne -1$
3	$\int \frac{1}{x}dx$	$\ln \left|x\right|+C$
4	$\int \frac{1}{a^2+x^2}dx$	$\frac{1}{a}\arctan \frac{x}{a}+C$
5	$\int \frac{1}{\sqrt{a^2-x^2}}dx$	$\arcsin \frac{x}{a}+C$
6	$\int \frac{-1}{\sqrt{a^2-x^2}}dx$	$\arccos \frac{x}{a}+C$
7	$\int \frac{1}{x\sqrt{x^2-a^2}}dx$	$\frac{1}{a}\text{arcsec}\frac{|x|}{a}+C$
8	$\int \ln xdx$	$x\ln x-x+C$
9	$\int {\log }_{b}xdx$	$x{\log }_{b}x-x{\log }_{b}e+C$
10	$\int e^{x}dx$	$e^x+C$
11	$\int a^{x}dx$	$\frac{a^x}{\ln a}+C$
12	$\int \sin xdx$	$-\cos x+C$
13	$\int \cos xdx$	$\sin x+C$
14	$\int \tan xdx$	$-\ln |\cos x|+C$
15	$\int \cot xdx$	$\ln |\sin x|+C$
16	$\int \sec xdx$	$\ln |\sec x+\tan x|+C$
17	$\int \csc xdx$	$-\ln |\csc x+\cot x|+C$
18	$\int {\sec }^{2}xdx$	$\tan x+C$
19	$\int {\csc }^{2}xdx$	$-\cot x+C$
20	$\int \sec x\tan xdx$	$\sec x+C$
21	$\int \csc x\cot xdx$	$-\csc x+C$
22	$\int {\sin }^{2}xdx$	$\frac{1}{2}(x-\sin x\cos x)+C$
23	$\int {\cos }^{2}xdx$	$\frac{1}{2}(x+\sin x\cos x)+C$
24	$\int {\sin }^{n}xdx$	$-\frac{{\sin }^{n-1}x\cos x}{n}+\frac{n-1}{n}\int {\sin }^{n-2}xdx$
25	$\int {\cos }^{n}xdx$	$-\frac{{\cos }^{n-1}x\sin x}{n}+\frac{n-1}{n}\int {\cos }^{n-2}xdx$
26	$\int \arctan xdx$	$x\arctan x-\frac{1}{2}\ln |1+x^2|+C$
27	$\int \sinh xdx$	$\cosh x+C$
28	$\int \cosh xdx$	$\sinh x+C$
29	$\int \tanh xdx$	$\ln |\cosh x|+C$
30	$\int \text{csch}xdx$	$\ln |\tanh \frac{x}{2}|+C$
31	$\int \text{sech}xdx$	$\arctan (\sinh x)+C$
32	$\int \coth xdx$	$\ln |\sinh x|+C$

Definite Integral

$\underset{a}{\overset{b}{\int }}f(x),dx$

$\underset{a}{\overset{b}{\int }}f(x)dx=F(b)-F(a).$

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

$\frac{d}{dx}\underset{a}{\overset{x}{\int }}f(t)dt=f(x).$

References

1. Integral