Integration by substitution

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2020 Mathematics Subject Classification: Primary: 26A06 [MSN][ZBL]

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 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 [S] and to Area formula.

The analogue of \eqref{e:change_of_var} for the 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.


[Ap] T.M. Apostol, "Mathematical analysis". Second edition. Addison-Wesley (1974) MR0344384 Zbl 0309.2600
[IlPo] V.A. Il'in, E.G. Poznyak, "Fundamentals of mathematical analysis" , 1–2 , MIR (1982) (Translated from Russian) MR0687827 Zbl 0138.2730
[Ku] L.D. Kudryavtsev, "Mathematical analysis" , 1 , Moscow (1973) (In Russian) MR0619214 Zbl 0703.26001
[Ni] S.M. Nikol'skii, "A course of mathematical analysis" , 1 , MIR (1977) (Translated from Russian) MR0466435 Zbl 0384.00004
[Ru] W. Rudin, "Principles of mathematical analysis", Third edition, McGraw-Hill (1976) MR038502 Zbl 0346.2600
[Ru] K.R. Stromberg, "Introduction to classical real analysis" , Wadsworth (1981)
How to Cite This Entry:
Integration by substitution. Encyclopedia of Mathematics. URL:
This article was adapted from an original article by V.A. Il'in (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article