Engineering Tables/Properties of Integrals

Table of Properties of Integrals
	Rule	Conditions
1	$\int adx=ax$
2 Homogeneity	$\int af(x)dx=a\int f(x)dx$
3 Distributivity	$\int (f\pm g\pm h\pm \cdots )dx=\int fdx\pm \int gdx\pm \int hdx\pm \cdots$
4 Integration by Parts	$\underset{a}{\overset{b}{\int }}f{g}^{\prime }dx=[fg{]}_a^b-\underset{a}{\overset{b}{\int }}g{f}^{\prime }dx$
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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