If \(f(x)=\int_{0}^{x}\displaystyle t\sin t\,dt\), then \(f'(x)\) is
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → \(x\sin x\)
By the first fundamental theorem of calculus, if \(f(x)=\int_{0}^{x} g(t)\,dt\) with g continuous, then \(f'(x) = g(x)\).
Here \(g(t) = t\sin t\), so
\(f'(x) = x\sin x\)
No integration is needed: the variable x simply replaces t in the integrand.