Question

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

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

Answer 1

Answer: (A)

$e^x\tan x+C$

We can write $f\left(x\right)=\frac{2si{n}^{3}x+2si{n}^{2}x-sinx-1-2si{n}^{3}x-si{n}^{2}x+2sinx+1}{cosx(1+sinx)}$
$=\frac{sinx(1+sinx)}{cosx(1+sinx)}=tanx$
We know,$\int e^x\left(f\right(x)+f^{\prime }(x\left)\right)=e^{x}f\left(x\right)+c=e^{x}tanx+c$