Integrating inverse functions using integration by parts

[14sohngenk]

Problem #12, it is just absolutely blowing my mind. Any help at all would be greatly appreciated. Bear with me here- a) Let y=f^-1(x) be the inverse function of f. Use integration by parts to derive the formula int(f^-1(x)dx) = xf^-1(x) - int(f(y)dy) b) Use the formula in part a to find the integral int(arcsin(x)dx c) Use the formula to find the area under the graph of y=ln(x), [1,e] Since I stunk at explaining this I'll try to attached a picture too. Thanks!

 

[HallsofIvy]

The problem says (1) let $\displaystyle y= f^{-1}(x)$ and (2) use integration by parts.
Okay, with $\displaystyle \int f^{-1}(x)dx= \int y(x)dx$, let u= y and dv= dx. Then $\displaystyle \int ydx= xy- \int x dy$. What is that in terms of $\displaystyle f^{-1}$?