Proof of indefinite integral of inverse function 
Assume that $f$ has a inverse on its domain.
a. Let $y = f^{-1}(x)$ and show that
      $\int f^{-1}(x)dx=\int yf'(y)dy$
b. Use part a to show that
$\int f^{-1}(x)dx= yf(y)-\int yf(y)dy$
c. Use part b to evaluate $\int ln(x)dx$
d. Use part b to evaluate $\int sin^{-1}(x)dx$
e. Use part b to evaluate $\int tan^{-1}(x)dx$

Parts c through e (and perhaps even b) seem fairly straightforward, but I'm stumped on how to prove a. I have a feeling I'm to use integration by parts, but every time I try I hit a dead end. 
Any help would be appreciated.

$f^{-1}(x)=y \implies f(y)=x$. Differentiating both sides of $f(y)=x$ gives $ f'(y) \cdot \frac{dy}{dx}=1 \implies f'(y) dy=dx$ Now just substitute. 
