How to find the antiderivative and derivative of an integral?

Question

I'm having trouble with finding $f'(x)$ and $f(\pi)$ for the following function. I think it has something do do with factoring out the $t$, but I'm not sure, can anyone please help me out?

Suppose that $f$ is a continuous function and that for $x > 0$.

$$\int_{0}^{x}tf(t)\;dt = x\sin(x)+\cos(x)-1$$

Answer 1

To find $f'(x)$: You need to find $f(x)$ first, for which you should differentiate the given equation twice. For the first differentiation, apply Leibniz rule on the LHS.

To find $f(\pi)$: You just need to put $x=\pi$ in the equation for $f(x)$ you had found in the previous step.

Answer 2

By using fundamental theorem of calculus:$$\frac{d}{dx}\left(\int_{0}^{x}tf(t)\;dt\right) = \frac{d}{dx}\left(x\sin(x)+\cos(x)-1\right)$$

$$xf(x) = \frac{d}{dx}\left(x\sin(x)+\cos(x)-1\right)$$

Hopefully you can take it from there.