Engineering Tables/Properties of Integrals

From Wikibooks, open books for an open world
Jump to: navigation, search
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.