QJEEM23Ma9d11: JEE Main 2023 Mathematics - Class 12 JEEM23Ma9d11 Solution: If and , then is equal to :

Question

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

Accepted Answer

\begin{aligned}&amp; \text { Given, } I(x)=\int e^{\sin ^2 x}(\cos x \sin 2 x-\sin x) d x \\\\&amp; =\int e^{\sin ^2 x} \cdot \cos x \cdot \sin 2 x d x-\int \sin x e^{\sin ^2 x} d x \\\\&amp; =\int \frac{\cos x}{\mathrm{I}} \cdot \frac{e^{\sin ^2 x} \cdot \sin 2 x}{\mathrm{II}} d x-\int \sin x \cdot e^{\sin ^2 x} d x \\\\&amp; =\cos x \cdot e^{\sin ^2 x}-\int(-\sin x) e^{\sin ^2 x} d x-\int \sin x e^{\sin ^2 x} d x \\\\&amp; =\cos x \cdot e^{\sin ^2 x}+\int \sin x e^{\sin ^2 x} \cdot d x-\int \sin x \cdot e^{\sin ^2 x} d x+C\end{aligned}

I(x)=e^{\sin ^2 x} \cdot \cos x+C

Given,

I(0)=1 \Rightarrow C=0

So,

I(x)=e^{\sin ^2 x} \cdot \cos x

\Rightarrow I(\pi / 3)=e^{3 / 4} \cdot \frac{1}{2}=\frac{e^{3 / 4}}{2}

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