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?

Hint:
By parts (integrating an implicit factor $1$),
$$I=\int\sqrt{25-x^2}dx=x\sqrt{25-x^2}+\int\frac{x^2}{\sqrt{25-x^2}}dx=x\sqrt{25-x^2}+\int\frac{25-(25-x^2)}{\sqrt{25-x^2}}dx.$$
Then
$$2I=x\sqrt{25-x^2}+\int\frac{25\,dx}{\sqrt{25-x^2}}.$$
After rescaling the variable, you will recognize the derivative of the $\arcsin$.
