Question

$\int e^{{\sin }^{2}x}\sin x(\cos x+{\cos }^{3}x)dx$ is equal to
(where $C$ is constant of integration)

• A. $\frac{1}{2}{e}^{{\sin }^{2}x}(3-{\sin }^{2}x)+C$
• B. $\frac{1}{2}{e}^{{\sin }^{2}x}(1-\frac{1}{2}{\cos }^{2}x)+C$
• C. $e^{{\sin }^{2}x}(3{\cos }^{2}x+2{\sin }^{2}x)+C$
• D. $e^{{\sin }^{2}x}(2{\cos }^{2}x+3{\sin }^{2}x)+C$

Answer 1

The correct option is A $\frac{1}{2}{e}^{{\sin }^{2}x}(3-{\sin }^{2}x)+C$

$I=\int e^{{\sin }^{2}x}\sin x(\cos x+{\cos }^{3}x)dx$
$=\frac{1}{2}\int e^{{\sin }^{2}x}\sin 2x(2-{\sin }^{2}x)dx$
Put ${\sin }^{2}x=t\Rightarrow \sin 2xdx=dt$
$\therefore I=\frac{1}{2}\int e^t(2-t)dt$
$=e^t-\frac{1}{2}\int t{e}^{t}dt$
$=e^t-\frac{1}{2}[t{e}^t-e^t]+C$
$=\frac{3}{2}{e}^t-\frac{1}{2}t{e}^t+C$
$=\frac{1}{2}(3-{\sin }^{2}x)e^{{\sin }^{2}x}+C$