Question

$\int (secx{)}^m(ta{n}^{3}x+tanx)dx$ is equal to

• A. $se{c}^{m+2}x+C$
• B. $se{c}^{m-2}x+C$
• C. $\frac{se{c}^{m+2}x}{m+2}+C$
• D. $\frac{ta{n}^{m+2}x}{m+2}+C$
• E. $\frac{se{c}^{m+1}x}{m-1}+C$

Answer 1

Answer: (C)

Let $I=\int (\sec x{)}^m\left({\tan }^{3}x+\tan x\right)dx$
$=\int {\sec }^{m}x\tan x\left({\tan }^{2}x+1\right)dx$
$=\int {\sec }^{m+1}x\cdot (\sec x\tan x)dx$
Put sec $x=t\Rightarrow \sec x\tan xdx=dt=\int t^{m+1}dt$
$=\frac{{\sec }^{m+2}x}{m+2}+C$