# Engineering Tables/Properties of Integrals

Table of Properties of Integrals
Rule Conditions
1 ${\displaystyle \int a\,dx=ax}$
2
Homogeneity
${\displaystyle \int af(x)\,dx=a\int f(x)\,dx}$
3
Distributivity
${\displaystyle \int {\left(f\pm g\pm h\pm \cdots \right)\,dx}=\int f\,dx\pm \int g\,dx\pm \int h\,dx\pm \cdots }$
4
Integration by Parts
${\displaystyle \int _{a}^{b}fg'\,dx={\bigl [}fg{\bigr ]}_{a}^{b}-\int _{a}^{b}gf'\,dx}$
4
General Integration by Parts
${\displaystyle \int f^{(n)}g\,dx=f^{(n-1)}g'-f^{(n-2)}g''+\dots +(-1)^{n}\int fg^{(n)}\,dx}$
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:
1. f, g, h are functions of x
2. a, n are constants.
3. The constant of integration, C has been omitted from this table. It should be included in the working of the equation if applicable.