If I(x) = e^^2x( x2x- x),dx and I(0)=1, then I(3) is equal to:

Question

If $I(x) = \displaystyle\int e^{\sin^2x}\big(\cos x\sin2x-\sin x\big)\,dx$ and $I(0)=1$, then $I\left(\dfrac\pi3\right)$ is equal to:

Accepted Answer

Since $\sin2x=2\sin x\cos x$, write $\cos x\sin2x - \sin x = 2\sin x\cos^2x-\sin x$. Splitting $I$ into two parts and integrating by parts on $\int e^{\sin^2x}\sin2x\cos x\,dx$ (with $\sin^2x=t$), the $\sin x$ terms cancel, leaving

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

Using $I(0)=1$: $e^0\cdot1+C=1 \implies C=0$.

$$I\left(\frac\pi3\right) = e^{\sin^2(\pi/3)}\cos\frac\pi3 = e^{3/4}\cdot\frac12 = \frac12 e^{3/4}$$

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