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Homesubstitution for integration
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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
$\int\frac{x\,dx}{1+x^{4}}=\frac{1}{2}\int\frac{dt}{1+t^{2}}=\frac{1}{2}\arctan t% +C=\frac{1}{2}\arctan x^{2}+C.$ 
Related:
IntegrationOfRationalFunctionOfSineAndCosine, IntegrationOfFractionPowerExpressions, ChangeOfVariableInDefiniteIntegral
Synonym:
variable changing for integration, integration by substitution, substitution rule
Type of Math Object:
Theorem
Major Section:
Reference
Parent:
Groups audience:
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