View source for Integration by substitution

[[Category:Analysis]]
{{MSC|26A06}}
{{TEX|done}}

One of the methods for calculating an integral in one real variable. It consists in transforming the integral by transition to another variable of integration. For the [[Definite integral|definite integral]] the formula is
\begin{equation}\label{e:change_of_var}
\int_a^b f(x)\, dx = \int_{\alpha}^\beta f (\phi (x)) \phi' (x)\, dx\, .
\end{equation}
This formula holds, for instance, under the following assumptions:

* $f:[a,b]\to \mathbb R$ is continuous;
* $\phi: [\alpha, \beta]\to [a,b]$ is continuously differentiable;
* $\phi (\alpha) = a$ and $\phi (\beta)=b$. 

However, these assumptions can be relaxed considerably: we refer to {{Cite|S}} and to [[Area formula]].

The analogue of \eqref{e:change_of_var} for the [[Indefinite integral|indefinite integral]] is the assertion that, if $F$ is a primitive of $f$, then $F\circ \phi$ is a primitive of $(f\circ \phi) \phi'$, which is an obvious corollary of the chain rule.

The formula \eqref{e:change_of_var} can be generalized to integrals in more than one variable: we refer to [[Change of variables in an integral]] and [[Area formula]].

====References====
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|valign="top"|{{Ref|Ap}}||valign="top"|  T.M. Apostol,   "Mathematical analysis". Second edition. Addison-Wesley   (1974) {{MR|0344384}} {{ZBL|0309.2600}} 
|-
|valign="top"|{{Ref|IlPo}}||valign="top"|  V.A. Il'in,   E.G. Poznyak,   "Fundamentals of mathematical analysis" ,  '''1–2''' , MIR  (1982)  (Translated from Russian) {{MR|0687827}}   {{ZBL|0138.2730}} 
|-
|valign="top"|{{Ref|Ku}}||valign="top"|  L.D. Kudryavtsev,   "Mathematical analysis" , '''1''' , Moscow  (1973)   (In Russian) {{MR|0619214}} {{ZBL|0703.26001}}
|-
|valign="top"|{{Ref|Ni}}||valign="top"|  S.M. Nikol'skii,   "A course of mathematical analysis" , '''1''' , MIR   (1977)  (Translated from Russian) {{MR|0466435}} {{ZBL|0384.00004}}
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|valign="top"|{{Ref|Ru}}||valign="top"|  W. Rudin, "Principles of mathematical analysis", Third edition,  McGraw-Hill (1976) {{MR|038502}} {{ZBL|0346.2600}}  
|-
|valign="top"|{{Ref|Ru}}||valign="top"| K.R. Stromberg,   "Introduction to classical real analysis" , Wadsworth  (1981)
|-
|}