What are your favorite applications of integration by parts?

(The answers can be as lowbrow or highbrow as you wish. I'd just like to get a bunch of these in one place!)

Thanks for your contributions, in advance! 

I always liked the derivation of Taylor's formula with error term:

$$\begin{array}{rl}
f(x) &= f(0) + \int_0^x f'(x-t) \,dt\\
&= f(0) + xf'(0) + \int_0^x tf''(x-t)\,dt\\
&= f(0) + xf'(0) + \frac{x^2}2f''(0) + \int_0^x \frac{t^2}2 f'''(x-t)\,dt
\end{array}$$

and so on. Using the mean value theorem on the final term readily gives the Cauchy form for the remainder.
