If $I(x)=\int e^{\sin^{2}x}(\cos x \sin 2x-\sin x)dx$ and $I(0)=1$, then $I\left(\frac{\pi}{3}\right)$ is equal to

(C) $\frac{1}{2}e^{\frac{3}{4}}$

(A) $-\frac{1}{2}e^{\frac{3}{4}}$

JEE Main · 2023 · 10 Apr 2023 · Shift 1 · Indefinite Integrals

I (x) = \int e^{s i n^{2} x} (cos x sin 2 x - sin x) d x

and

I (0) = 1

, then

I (\frac{π}{3})

is equal to

Answer: (C)

Since

sin 2 x = 2 sin x cos x

, the integrand is

e^{s i n^{2} x} (2 sin x cos^{2} x - sin x)

. Now

\frac{d}{d x} (e^{s i n^{2} x} cos x) = e^{s i n^{2} x} (2 sin x cos x) cos x - e^{s i n^{2} x} sin x = e^{s i n^{2} x} (2 sin x cos^{2} x - sin x)

, which is exactly the integrand. Therefore

I (x) = e^{s i n^{2} x} cos x + C

. The condition

I (0) = 1

gives

e^{0} \cdot 1 + C = 1

, so

C = 0

. Hence

I (\frac{π}{3}) = e^{s i n^{2} (π /3)} cos \frac{π}{3} = e^{3/4} \cdot \frac{1}{2} = \frac{1}{2} e^{3/4}

JEE Main · 2023 · 10 Apr 2023 · Shift 1 · Single correct option