If $f(x)=\displaystyle\int_0^{x} t\sin t\,dt$, then $f'(x)$ is
A$\cos x+x\sin x$
B$x\sin x$
C$x\cos x$
D$\sin x+x\cos x$
Answer & Solution
Correct answer: B. $x\sin x$
1. By the First Fundamental Theorem, if $f(x)=\int_a^x g(t)\,dt$ then $f'(x)=g(x)$.
2. Here $g(t)=t\sin t$.
3. So $f'(x)=x\sin x$.
4. Options A and D mistakenly differentiate the integrand.
_Source: NCERT Class 12 Mathematics Ch 7 "Integrals", p.341_
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