Engineering Tables/Properties of Integrals

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.

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