Question 8.T.13: (First Substitution Rule) Suppose φ is differentiable on [a,...
(First Substitution Rule)
Suppose φ is differentiable on [a, b] and its derivative φ^{\prime} is continuous.
If f is continuous on the range of φ, then
\int_{a}^{b}{f (\varphi (t)) \varphi^{\prime}(t)} dt = \int_{\varphi(a)}^{\varphi(b)} {f (x)} dx (8.18)
Let J be the interval whose end-points are φ (a) and φ (b) .
Since φ is continuous, its range I is an interval, and since both φ (a) and φ (b) lie in I , I ⊇ J. f is therefore continuous on J and the integral on the right-hand side of (8.18) exists. The continuity of f, φ, and φ^{\prime} ensures the existence of the integral on the left-hand side of (8.18). All that remains is to prove the equality.
Let us define F : I → \mathbb{R} and G : [a, b] → \mathbb{R} as follows:
F (y) = \int_{φ(a)}^{y}{f (x)} dx, G = F \circ φ.
From the chain rule and Theorem 8.11, we have
G^{\prime} (t) = F^{\prime} (φ (t)) φ^{\prime} (t) = f (φ (t)) φ^{\prime} (t) , t ∈ [a, b] .
Noting that G (a) = 0, and using Theorem 8.12, we see that
G (b) = \int_{a}^{b}{(φ (t)) φ^{\prime} (t)} dt,
which proves (8.18).