# 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
$\int_a^b f g'\,dx = \left[ f g \right]_{a}^{b} - \int_a^b g f' \,dx$
4
General Integration by Parts
$\int f^{(n)} g \,dx = f^{(n-1)}g' - f^{(n-2)}g'' + \ldots + (-1)^n \int f g^{(n)} \,dx$
5 $\int f(ax) \,dx = \frac{1}{a} \int f(x) \, dx$
6
Substitution Rule
$\int g \{ f (x) \} \,dx = \int g(u) \frac{dx}{du} \, du = \int \frac{g(u)}{f'(x) } \,du$ $u= f(x)\,$
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:
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.