# Question 8.T.14: (Second Substitution Rule) Let the function φ: [a, b] → R ha...

(Second Substitution Rule)

Let the function φ : [a, b] → \mathbb{R} have a continuous derivative that does not vanish anywhere in (a, b) . If f is continuous on the range of φ, and ψ is the inverse of φ, then

\int_{a}^{b}{f (φ (t)) dt} = \int_{φ(a)}^{φ(b)}{f(x) ψ^{\prime} (x) dx}. (8.19)

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