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

Question

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

Accepted Answer

$\begin{gathered} I(x)=\int {e}^{{\sin }^{2}x}(\cos x\sin 2x-\sin x)dx \ =\int {e}^{{\sin }^{2}x}\cdot \cos x\cdot \sin 2xdx-\int \sin x{e}^{{\sin }^{2}x}dx \ =\int \frac{\cos x}{I}\cdot \frac{{e}^{{\sin }^{2}x}\cdot \sin 2x}{II}dx-\int \sin x\cdot {e}^{{\sin }^{2}x}dx \ =\cos x\cdot {e}^{{\sin }^{2}x}-\int (-\sin x){e}^{{\sin }^{2}x}dx-\int \sin x{e}^{{\sin }^{2}x}dx \ =\cos x\cdot {e}^{{\sin }^{2}x}+\int \sin x{e}^{{\sin }^{2}x}\cdot dx-\int \sin x\cdot {e}^{{\sin }^{2}x}dx+C \end{gathered}$

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

Given, $I(0)=1\Rightarrow C=0$

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

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

Solutions · 1