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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?

Crawl
  • 51

1 Answers1

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From where you left,

$$\int_a^b f(x) dx + \int_a^b xf’(x) dx = \int_a^b (xf(x))’ \ dx = \left[ xf(x)\right]_a^b = bf(b)-af(a) $$

Vishu
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