STD 12 - 7.2 definite integral — JEE कक्षा 11 साइन्स — Question

Integrating by parts, we obtain

$f(x)=t \int_{0}^{x} \sin t d t-\int_{0}^{x}\left\{\left(\frac{d}{d t} t\right) \int \sin t \, d t\right\} d t$

$=[t(-\cos t)]_{0}^{x}-\int_{0}^{x}(-\cot t) d t$

$=[-t \cos t+\sin t]_{0}^{x}$

$=-x \cos x+\sin x$

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

$=x \sin x-\cos x+\cos x$

$=x \sin x$

Hence, the correct Answer is $B$