Chapter 7.3, Problem 52E

To determine

To show that ${\displaystyle \int f^{-1}(x)dx=x{f}^{-1}(x)-{\displaystyle \int x\frac{d}{dx}\left(f^{-1}(x)\right)dx}}$ .

Explanation of Solution

Given information: ${\displaystyle \int f^{-1}(x)dx}$

Formula used: ${\displaystyle \int uvdx=u{\displaystyle \int vdx-{\displaystyle \int \left(u^{\prime }{\displaystyle \int vdx}\right)}}}dx$

Calculation :

Integrating by parts, the given integral becomes:

  ${\displaystyle \int f^{-1}(x).1dx}=f^{-1}(x).1-{\displaystyle \int \frac{d}{dx}\left(f^{-1}(x)\right).xdx}$

Which is the required expression