substitution for integration

For determining the antiderivative F(x) of a given real function f(x) in a “closed formMathworldPlanetmathPlanetmath”, i.e. for integrating f(x), the result is often obtained by using the




and  x=x(t)  is a differentiable function, then

F(x(t))=f(x(t))x(t)𝑑t+c. (1)

Proof.   By virtue of the chain ruleMathworldPlanetmath,


and according to the supposition, F(x)=f(x).  Thus we get the claimed equation (1).


  • The expression x(t)dt in (1) may be understood as the differentialMathworldPlanetmath of x(t).

  • For returning to the original variable x, the inverse functiont=t(x)  of x(t) must be substituted to F(x(t)).

Example.   For integrating xdx1+x4 we take  x2=t  as a new variable.  Then,  2xdx=dt, xdx=dt2, and we get

Title substitution for integration
Canonical name SubstitutionForIntegration
Date of creation 2013-03-22 14:33:38
Last modified on 2013-03-22 14:33:38
Owner pahio (2872)
Last modified by pahio (2872)
Numerical id 21
Author pahio (2872)
Entry type Theorem
Classification msc 26A36
Synonym variable changing for integration
Synonym integration by substitution
Synonym substitution rule
Related topic IntegrationOfRationalFunctionOfSineAndCosine
Related topic IntegrationOfFractionPowerExpressions
Related topic ChangeOfVariableInDefiniteIntegral