Q:

The Fundamental Theorem of Calculus

So I have this math problem and I don't know how to solve it...
Find $f(4)$ $$\int_0^{x^2} f(t) \, dt= x\cos(\pi x)$$
This is what I came up with but I have no idea if it is right or what to do next. $$F(x^2)-F(0)=x(\cos\pi x)$$ 

A:

Iulius Caesar said: DERIVE AND RULE.
$$\cos(\pi x)-\pi x\sin(\pi x)=\frac{d}{dx}\left(x\cos(\pi x)\right) =\frac{d}{dx} \int_0^{x^2}f(t)dt=2x f(x^2).$$
Hence, $\cos(\pi x)-\pi x\sin(\pi x)=2x f(x^2)$, and you are done:
$$f(4)=1/4.$$
EDIT:
Start with $2xf(x^2)=\cos(\pi x)-\pi x \sin(\pi x)$ and take $x=2$. So $$4f(4)=\cos (2\pi)-2\pi\sin(2\pi)=1,$$ so $f(4)=1/4$.

