Substitution for definite integrals
What are some important applications of change of variables in definite integration, other than symbolic evaluation?
Here are a handful:
To give a computational (as opposed to geometric) proof that $$ \int_{-a}^{a} f(x)\, dx = \begin{cases} 0 & \text{if $f$ is odd}; \\ 2\displaystyle\int_{0}^{a} f(x)\, dx & \text{if $f$ is even.} \end{cases} $$
To show that if $a$ and $b$ are positive real numbers, then $$ \int_{a}^{ab} \frac{1}{t}\, dt = \int_{1}^{b} \frac{1}{t}\, dt, $$ which, of course, is the key to proving $\log(ab) = \log(a) + \log(b)$.
To prove that if $f$ is continuous on $[-1, 1]$, then $$ \int_{0}^{2\pi} f(\cos \theta) \sin\theta\, d\theta = \int_{0}^{2\pi} f(\sin \theta) \cos\theta\, d\theta = 0. $$