# Engineering Tables/Properties of Integrals

Table of Properties of Integrals
Rule Conditions
1 $\int a\,dx=ax$ 2
Homogeneity
$\int af(x)\,dx=a\int f(x)\,dx$ 3
Distributivity
$\int {\left(f\pm g\pm h\pm \cdots \right)\,dx}=\int f\,dx\pm \int g\,dx\pm \int h\,dx\pm \cdots$ 4
Integration by Parts
$\int _{a}^{b}fg'\,dx={\bigl [}fg{\bigr ]}_{a}^{b}-\int _{a}^{b}gf'\,dx$ 4
General Integration by Parts
$\int f^{(n)}g\,dx=f^{(n-1)}g'-f^{(n-2)}g''+\dots +(-1)^{n}\int fg^{(n)}\,dx$ 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:
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.