Striking applications of integration by parts 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:

$$\begin{array}{rl}
f(x) &amp;= f(0) + \int_0^x f&#39;(x-t) \,dt\\
&amp;= f(0) + xf&#39;(0) + \int_0^x tf&#39;&#39;(x-t)\,dt\\
&amp;= f(0) + xf&#39;(0) + \frac{x^2}2f&#39;&#39;(0) + \int_0^x \frac{t^2}2 f&#39;&#39;&#39;(x-t)\,dt
\end{array}$$

and so on. Using the mean value theorem on the final term readily gives the Cauchy form for the remainder.
Integration by parts shows that (modulo a constant) the Fourier transform interchanges differentiation and multiplication by the variable:
$\begin{align*}
f'(x) \rightarrow \widehat{f'}(\xi) &amp; = \int_{\mathbb{R}} f'(x)e^{-2 \pi i x \xi}dx\\
 &amp; = f(x)e^{-2 \pi i x \xi}|_{-\infty}^{\infty} - \int_{\mathbb{R}} f(x) e^{-2 \pi i x \xi} (-2 \pi i \xi) dx \\
&amp; = (2\pi i \xi) \widehat{f}(\xi)
\end{align*}$
where $f(x)e^{-2 \pi i x \xi}|_{-\infty}^{\infty}$ vanishes if $f$ decays fast.