# substitution for integration

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

###### Theorem.

If

 $\int f(x)\,dx=F(x)+C$

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

 $\displaystyle F(x(t))=\int f(x(t))\,x^{\prime}(t)\,dt+c.$ (1)

Proof.   By virtue of the chain rule,

 $\frac{d}{dt}F(x(t))=F^{\prime}(x(t))\cdot x^{\prime}(t),$

and according to the supposition, $F^{\prime}(x)=f(x)$.  Thus we get the claimed equation (1).

Remarks.

• The expression $x^{\prime}(t)\,dt$ in (1) may be understood as the differential of $x(t)$.

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

Example.   For integrating $\int\frac{x\,dx}{1+x^{4}}$ we take  $x^{2}=t$  as a new variable.  Then,  $2x\,dx=dt$, $x\,dx=\frac{dt}{2}$, and we get

 $\int\frac{x\,dx}{1+x^{4}}=\frac{1}{2}\int\frac{dt}{1+t^{2}}=\frac{1}{2}\arctan t% +C=\frac{1}{2}\arctan x^{2}+C.$
 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