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! 

There are a couple of applications in PDEs that I am quite fond of. As well as verifying that the Laplace operator $-\Delta$ is positive on $L^2$, I like the application of integration by parts in the energy method to prove uniqueness.  
Suppose $U$ is an open, bounded and connected subset of $\mathbb{R}^n$. Introduce the BVP
\begin{equation*}
-\Delta u=f~\text{in}~U
\end{equation*}
with initial position $f$ on the boundary $\partial U$. Suppose $v\in C^2(\overline{U})$ and set $w:=u-v$ such that we can establish a homogeneous form of our equation. Then an application of integration by parts gives us
\begin{equation*}
0=-\int_U w\Delta wdx=\int_U \nabla w\cdot \nabla wdx-\int_{\partial U}w\frac{\partial w}{\partial\nu}dS=\int_U|\nabla w|^2dx
\end{equation*}
with outward normal $\nu$ of the set $U$. By establishing that $\nabla w=0$, we can then conclude uniqueness of the solution in $U$.
