`intsec^(4)xtanxdx=?`

Options

A. `(1)/(2)sec^(2)x+(1)/(4)sec^(4)x+C`
B. `(1)/(2)tan^(2)x+(1)/(4)tan^(4)x+C`
C. `(1)/(2)secx+log|secx+tanx|+C`
D. none of these

Answer

B

Explanation

`I=intsec^(2)x*sec^(2)x*tanxdx`.
`=intsec^(2)x(1+tan^(2)x)tanxdx=int(1+t^(2))tdt`, where tan x=t
`int(t+t^(3))dt=(t^(2))/(2)+(t^(4))/(4)+C=(1)/(2)tan^(2)x+(1)/(4)tan^(4)x+C`.