Question

$\int e^x\left[\frac{sinx+cosx}{1-si{n}^{2}x}\right]dx$ is

• A. $(e^x.\cos ecx)+c$
• B. $(e^x\cot x)+c$
• C. $(e^x.\sec x)+c$
• D. $(e^x\tan x)+c$

Answer 1

Answer: (C)

$I=\int e^x\left(\frac{\sin x+\cos x}{1-{\sin }^{2}x}\right)dx$
$I=\int e^x\left(\frac{\sin x+\cos x}{{\cos }^{2}x}\right)dx$
$I=\int e^x\tan x\cdot \sec xdx+\int \frac{e^x}{II}\cdot \frac{\sec }{I}xdx$
$I=\int e^x\tan x\cdot \sec xdx+\{\sec x\cdot e^x$
$-\int \sec x\cdot \tan x{e}^{x}dx\}$
$I=\int e^x\tan x\cdot \sec xdx+e^x\sec x$
$-\int e^x\tan x\cdot \sec xdx$
$I\equiv e^x\cdot \sec x+C$