Real functions Riemann and related integrals Theorem

Integration by substitution

Builds on Chain rule Second fundamental theorem of calculus The integral of a continuous function is an antiderivative Continuously differentiable function

Statement

Let $g \colon [a, b] \to \mathbb{R}$ be continuously differentiable, and let $f$ be continuous on an interval that contains $g([a, b])$ . Then $\int_a^b f(g(x)) \, g'(x) \, dx = \int_{g(a)}^{g(b)} f(u) \, du.$

Remarks

Let $F$ be an antiderivative of $f$ . By the chain rule, $F \circ g$ is an antiderivative of $(f \circ g) \, g'$ . Both sides equal $F(g(b)) - F(g(a))$ . The function $g$ need not be injective. For example, with $u = x^2$ , $\int_0^1 2x \cos(x^2) \, dx = \int_0^1 \cos u \, du = \sin 1$ . Mathlib states it as intervalIntegral.integral_comp_mul_deriv.

Reading path

To understand this page from the ground up, read these 52 pages in order. The foundations come first.

Limit point of a set definition
Local maximum definition
Partition of an interval definition
Sequence definition
Limit of a function definition
Local minimum definition
Mesh of a partition definition
Riemann sum definition
Real sequence definition
Subsequence definition
Continuity at a point definition
Derivative definition
Riemann integrable function definition
Bounded sequence definition
Limit of a sequence definition
Monotone sequence definition
Continuous function definition
Epsilon-delta definition of continuity statement
Antiderivative definition
Differentiable function definition
Fermat's theorem on stationary points statement
Riemann integral definition
Upper and lower Darboux sums definition
Convergent sequence definition
Monotone convergence theorem for sequences statement
Every sequence has a monotone subsequence statement
Intermediate value theorem statement
Composition of continuous functions is continuous statement
Uniform continuity definition
Derivative function definition
Definite integral definition
Upper and lower Darboux integrals definition
Bolzano–Weierstrass theorem statement
Chain rule statement
Uniform continuity on a set definition
Boundedness theorem statement
Higher derivative definition
Monotonicity of the integral statement
Additivity of the integral over intervals statement
Darboux and Riemann integrals agree statement
Extreme value theorem statement
Continuously differentiable function definition
Continuous functions on a closed bounded interval are uniformly continuous statement
Riemann's criterion for integrability statement
Mean value theorem for integrals statement
Rolle's theorem statement
Continuous functions are Riemann integrable statement
Mean value theorem statement
First fundamental theorem of calculus statement
Second fundamental theorem of calculus statement
The integral of a continuous function is an antiderivative statement
Integration by substitution statement

Where this leads

2 pages build on this one.

Tag	`0D37QFPEZE`
URI	`m2://0D37QFPEZE`
Aliases	`change-of-variables-formula`
Kind	statement, theorem
Writing	W1 stated
MSC	26A42
Status	This copy of the site has no build data. Run `m2 status 0D37QFPEZE`.
Uses	Chain rule, Second fundamental theorem of calculus, The integral of a continuous function is an antiderivative, Continuously differentiable function
Same as	Wikidata Q1071270 Wikipedia Integration by substitution

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