Question

$\int e^{x/2}\sec 2x(1+4\tan 2x)dx$ is equal to

• A. $4e^{x/2}\sec 2x+c$
• B. $2e^{x/2}\sec 2x+c$
• C. $e^{x/2}\sec 2x+c$
• D. ${}^{\frac{1}{2}{e}^{x/2}\sec 2x+c}$

Answer 1

Answer: (B)

$2e^{x/2}\sec 2x+c$

$\int e^{x/2}sec2x(1+4tan2x)dx$
$\int e^{x/2}2(sec2x+4tan2x.sec2x)dx$
It is in the form of
$\int e^x\left(f\right(x)+f^1(x\left)\right)dx$
$=e^{x}f\left(x\right)+c$
$=2e^{x/2}sec2x+c$
$\int e^{x}sec2x=2e^{x/2}sec2x+c$