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Please solve RD Sharma class 12 chapter 19 Definite Integrals exercise Very short answer type question 33 maths textbook solution

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Answer: f^{\prime}(x)=x \sin x

Hint: you must know the rule of integration

Given: f(x)=\int_{0}^{x} t \sin t\; d t

Solution:  f(x)=\int_{0}^{x} t \sin t\; d t

=\left[t \int \sin t\; d t-\int\left\{\frac{d t}{d t} \int \sin t \; d t\right\} d t\right]_{0}^{x}

\begin{aligned} &=\left[t(-\cos t)+\int \cos t d t\right]_{0}^{x} \\\\ &=[-t \cos t+\sin t]_{0}^{x} \end{aligned}

=-x \cos x+\sin x-(-0 \cos 0+\sin 0)

\begin{aligned} &f(x)=\sin x-x \cos x \\\\ &f^{\prime}(x)=\cos x-(x(-\sin x)+\cos x) \end{aligned}

           \begin{aligned} &=\cos x+x \sin x-\cos x \\\\ &=x \sin x \end{aligned}


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