Mathematics - Indefinite Integration Question with Solution | TestHub

$I=4\int \sin x\cos x\left(\frac{1-{\tan }^{2}x}{1+{\tan }^{2}x}\right)e^{{\tan }^{2}x}dx=2\int \frac{(1-t)e^t}{(1+t{)}^3}dt,t={\tan }^{2}x$ $=-\frac{2{\mathrm{e}}^{\mathrm{t}}}{(1+\mathrm{t}{)}^2}+\mathrm{c}\therefore \varphi (\mathrm{x})=-2{\mathrm{e}}^{{\tan }^2\mathrm{x}}{\cos }^4\mathrm{x}$

Clearly $\varphi (\mathrm{x})$ is even function,

$\varphi (\pi /4)=-\frac{\mathrm{e}}{2},\varphi \left(\frac{3\pi }{4}\right)=-\frac{\mathrm{e}}{2}$