Question

If $f\left(x\right)={\int }_0^{x}tsintdt$, then write the value of $f^{\prime }\left(x\right)$.

Answer 1

$f\left(x\right)={\int }_0^{x}t\sin tdt$

Integrate by parts,

$f\left(x\right)={\left[t\right(-\cos t\left)\right]}_0^x-{\int }_0^{x}1.(-\cos t)dt$
$\Rightarrow f\left(x\right)=-(x\cos x-0)+(\sin t{)}_0^x$
$\Rightarrow f\left(x\right)=-x\cos x+\sin x$

$\Rightarrow f^{\prime }\left(x\right)=-x(-\sin x)-\cos x+\cos x$

$\Rightarrow f^{\prime }\left(x\right)=xsinx$