Question

If $y={\int }_0^{x^2}\sin \sqrt{t}dt$, then $\frac{dy}{dx}$ is equal to

• A. $\sin x$
• B. $\frac{\cos x}{2x}$
• C. $2x\sin x$
• D. None of the above

Answer 1

The correct option is C $2x\sin x$

According Leibniz integral rule-

$\frac{d}{dx}{\int }_{a\left(x\right)}^{b\left(x\right)}f\left(x\right)dx$

$=f\left(b\right(x\left)\right)\frac{d}{dx}b\left(x\right)-f\left(a\right(x\left)\right)\frac{d}{dx}a\left(x\right)$

$⟹\frac{dy}{dx}=\left(\frac{d}{dx}{x}^2\right)\left(\sin \sqrt{x^2}\right)$

$\frac{dy}{dx}=2x\sin x$

Hence, answer is option-(C).