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[parent] substitution for integration (Theorem)

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 1   If
$\displaystyle \int f(x)\,dx = F(x)+C$
and $ x = x(t)$ is a differentiable function, then
$\displaystyle F(x(t)) = \int f(x(t))\,x'(t)\,dt+c.$ (1)

Proof. By virtue of the chain rule,

$\displaystyle \frac{d}{dt}F(x(t)) = F'(x(t))\cdot x'(t),$
and according to the supposition, $ F'(x) = f(x)$. Thus we get the claimed equation (1).

Remarks.

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

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



"substitution for integration" is owned by pahio.
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See Also: integration of rational function of sine and cosine, integration of fraction power expressions

Other names:  variable changing for integration, integration by substitution

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Attachments:
integration of differential binomial (Theorem) by rspuzio
a lecture on integration by substitution (Feature) by alozano
A lecture on trigonometric integrals and trigonometric substitution (Feature) by alozano
Euler's substitutions for integration (Topic) by pahio
integration of fraction power expressions (Application) by pahio
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Cross-references: inverse function, variable, expression, equation, chain rule, proof, differentiable function, real function, antiderivative
There are 5 references to this entry.

This is version 17 of substitution for integration, born on 2004-08-27, modified 2008-05-06.
Object id is 6114, canonical name is SubstitutionForIntegration.
Accessed 4119 times total.

Classification:
AMS MSC: 26A36 (Real functions :: Functions of one variable :: Antidifferentiation)
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