Integration by substitution
change of variable in an integral
One of the methods for calculating an integral. It consists in transforming the integral by transition to another variable of integration. For the definite integral of a function of one variable the formula is
![]() | (1) |
It is true under the assumptions: is continuous on the interval
, which is the range of a function
that is defined and continuous, together with its first derivative
, on an interval
, and
,
.
The analogue of (1) for the indefinite integral is
![]() | (2) |
If is defined and differentiable on some segment
, while
has a primitive on the range of
, then
also has a primitive on the given segment, and (2) holds.
In the case of a multiple Riemann integral over a bounded closed -dimensional measurable region
(cf. Multiple integral), the analogue of (1) is
![]() | (3) |
Formula (3) holds under the following assumptions: the function is continuous in
; the transformation
,
, maps a region
in the space of variables
one-to-one onto
; the functions
have in
continuous first-order partial derivatives, and their Jacobian
does not vanish. Formula (3) holds under more general assumptions as well (it is not necessary to require that
be continuous in
, and the Jacobian may vanish on a set of
-dimensional measure zero).
References
[1] | V.A. Il'in, E.G. Poznyak, "Fundamentals of mathematical analysis" , 1–2 , MIR (1982) (Translated from Russian) |
[2] | L.D. Kudryavtsev, "Mathematical analysis" , 1–2 , Moscow (1970) (In Russian) |
[3] | S.M. Nikol'skii, "A course of mathematical analysis" , 1–2 , MIR (1977) (Translated from Russian) |
Comments
For very general hypothesis under which (1) and (3) hold for Lebesgue integrals (cf. Lebesgue integral) see [a1].
For additional references see also Improper integral.
References
[a1] | K.R. Stromberg, "Introduction to classical real analysis" , Wadsworth (1981) |
Integration by substitution. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Integration_by_substitution&oldid=28766