Suppose $f'$ is continuous and positive on $[a, b]$. Prove $\int_a^b f(x) dx + \int_{f(a)}^{f(b)} f^{-1} (y) dy = b f(b) - a f(a)$

Question

Suppose $f'$ is continuous and positive on $[a, b]$. Prove $$\int_a^b f(x) dx + \int_{f(a)}^{f(b)} f^{-1} (y) dy = b f(b) - a f(a)$$

I'm attempting to apply a change of variables but unsure where to start.

Answer 1

Since $f'$ is positive on $[a,b]$, the function $f$ is injective on $[a,b]$. Try the substitution $y = f(x)$ on the second integral, and then note your desired equality is a case of integration by parts (or product rule).