Chapter 8, Problem 9PS

Inverse Function and Area

(a) Let $y=f^{-1}(x)$ be the inverse function of f. Use integration by parts to derive the formula

${\displaystyle \int f^{-1}(x)dx=x{f}^{-1}(x)-{\displaystyle \int f(y)dy}}$.

(b) Use the formula in part (a) to find the integral

${\displaystyle \int \arcsin xdx}$.

(c) Use the formula in part (a) to find the area under the graph of $y=\ln x,1\le x\le r$ (see figure).