Home page for accesible maths Math 101 Chapter 5: Integration 5.22 Integration by substitution 5.24 Proof of integration by substitution theorem

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5.23 Integration by substitution for definite integrals

Integration by substitution

Let $u$ be a differentiable function of $x$ . Then the definite integrals satisfy

\int_{u(a)}^{u(b)}f(u)\,du=\int_{a}^{b}f(u(x)){{du}\over{dx}}dx.

This formula for definite integrals holds even when the natural order of the limits is reversed by the substitution. The rule 5.12(iii) shows how to change the order of the limits; use the following table to change limits.

x\qquad|\qquad a\qquad\quad\qquad b

u\qquad|\quad u(a)\qquad\qquad u(b)