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

$F(x(t))={\displaystyle \int f(x(t))x^{\prime }(t)𝑑t}+c.$ (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 $\int \frac{xdx}{1+x^4}$ we take  $x^2=t$  as a new variable.  Then,  $2xdx=dt$, $xdx=\frac{dt}{2}$, and we get

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

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