Derivative of a function and an integral

Question

For example, if I had an equation like

d/dx($x^2$(definite accumulated integral))

How do I solve this? would I just use FTOC II to normally solve the integral then multiply by $x^2$?

Or am I supposed to use product rule for derivatives?

Here is what I'm working on...

$$ \frac d {dx} \left (x^6(\int_{0}^{sinx} \sqrt{t} dt)\right )$$

Answer 1

Generally:$$ \frac d {dx} \left (x^6(\int_{0}^{sinx} f(t) dt)\right )=6x^5 \int_{0}^{sinx} f(t) dt+x^6 \frac d {dx} \int_{0}^{sinx} f(t) dt$$if we assume $F(x)=\int_{0}^{x}f(t)dt$ we have$$\frac d {dx} \int_{0}^{sinx} f(t) dt=\dfrac{d}{dx}F(\sin x)=\cos x f(\sin x)$$therefore$$\frac d {dx} \left (x^6(\int_{0}^{sinx} f(t) dt)\right )=6x^5 \int_{0}^{sinx} f(t) dt+x^6 \cos x f(\sin x)$$here we have$$\frac d {dx} \left (x^6(\int_{0}^{sinx} \sqrt t dt)\right )=4x^5\sin x\sqrt{\sin x}+x^6\cos x\sqrt{\sin x}$$