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

We are given $$I(x)=\int e^{\sin^2 x}\cos x\sin2x\sin x\,dx$$ with $$I(0)=1$$, and we wish to find $$I(\pi/3)$$. Using the identity $$\sin2x=2\sin x\cos x$$ simplifies the integrand to $$2e^{\sin^2 x}\sin^2 x\cos^2 x\,. $$ Substituting $$t=\sin^2 x$$ yields $$dt=2\sin x\cos x\,dx\,, $$ so the integral becomes $$\int e^t\,t\,dt\,. $$ An antiderivative of this is $$e^t(t-1)+C\,, $$ and the condition $$I(0)=1$$ determines the constant. Finally, since $$\sin^2\bigl(\pi/3\bigr)=3/4\,, $$ we obtain $$I(\pi/3)=\tfrac12e^{3/4}\,. $$ Therefore the correct answer is Option B: $$\dfrac12e^{3/4}\,. $$