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_
Related questions
$\displaystyle\int_2^3 \dfrac{x\,dx}{x^2+1}$ equals$\displaystyle\int_0^{2a} f(x)\,dx$ equals $2\displaystyle\int_0^{a} f(x)\,dx$ precisely wIf $f(a+b-x)=f(x)$, then $\displaystyle\int_a^b x\,f(x)\,dx$ is equal to$\displaystyle\int \frac{dx}{e^{x}+e^{-x}}$ is equal toA rational function $\dfrac{P(x)}{Q(x)}$ is called proper when$\displaystyle\int_0^{\pi/4} \tan x\,dx$ equals$\displaystyle\int_0^{1} x e^{x^2}\,dx$ equalsThe value of $\displaystyle\int_0^{\pi/2} \log\!\left(\dfrac{4+3 in x}{4+3\cos x}\right)dx