How can one use the chain rule to integrate? I am trying to calculate the anti-derivative of $y=\sqrt{25-x^2}$, for which I believe I may need the chain rule $\frac{dy}{dx} = \frac{dy}{du} \times \frac{du}{dx}$. How I would use it, however, is a different matter entirely.  
I used this website for a tutorial, however my answer of $\left[\frac{(200-8x^2)^\frac{3}{4}}{7}\right]_0^5$ is vastly different from the actual answer $\frac{25\pi}{4}$. How exactly would someone calculate the antiderivative of a function like this?

I lied in my comment - the substitution rule is what you need but it's not used in the conventional way. See, we usually sub $u = g(x)$ so we make a hard problem easier. However, we need something different here.
We start with $\int \sqrt{25 - x^2} \ dx$. Well, let's remember a trig identity: $1 - \sin^2 x = \cos^2 x.$ So what do we do? Here's a trick. Let $x = 5 \sin w$. Then $25 - x^2 = 25 - 25 \sin^2 x$. What's next? Hey, pull a 25 out and we get something nice!
Here's what we got:
\begin{align*}
\int \sqrt{25 - x^2} \ dx &= \int 5 \cos w \sqrt{25 - 25 \sin^2 w} \ dw \\
&= \int 25 \cos^2 w \ dw
\end{align*}
Now with the help of another trig identity (half angle), we'll be done! Can you take it from here, pal?
