Question

Let $f(x)=\frac{2 \sin ^2 x-1}{\cos x}+\frac{\cos x(2 \sin x+1)}{1+\sin x}$ then $\int e ^{ x }\left( f ( x )+ f ^{\prime}( x )\right) dx$ where $c$ is the constant of integeration)

• A. $ e^x \tan x+c$
• B. $e ^{ x } \cot x + c$
• C. $e ^{ x } \operatorname{cosec}^2 x + c$
• D. $e ^{ x } \sec ^2 x + c$

Answer 1

Answer: (A)

$\frac{\cos x(1+2 \sin x)}{1+\sin x}-\frac{\cos ^2 x-\sin ^2 x}{\cos x}$
$=\frac{\cos ^2 x(1+2 \sin x)-(1+\sin x)\left(\cos ^2 x-\sin ^2 x\right)}{\cos x(1+\sin x)}=\frac{\sin x \cos ^2 x+\sin ^3 x+\sin ^2 x}{\cos x(1+\sin x)} $
$=\frac{\sin x\left(\cos ^2 x+\sin x\right)+\sin ^2 x}{\cos x(1+\sin x)}=\tan x$