the question is as follows.
$f$ is differentiable on $[a,b]$ and $f'$ is continuous. And $\forall x\in[a,b]$, f'(x) is not 0. Then, show that $$\int_a^b f(x)\,dx+ \int_{f(a)}^{f(b)} f^{-1}(x) \,dx = bf(b)-af(a)$$
I take the $$\int_{f(a)}^{f(b)} f^{-1}(x) \,dx = \int_a^b f^{-1}(f(x))f'(x) \,dx$$
I stuck here. How can i solve?
From where you left,
$$\int_a^b f(x) dx + \int_a^b xf’(x) dx = \int_a^b (xf(x))’ \ dx = \left[ xf(x)\right]_a^b = bf(b)-af(a) $$