`inttan^5xdx`

Options

A. `(1)/(6)tan^(6)x+C`
B. `(1)/(4)tan^(4)x+(1)/(2)tan^(2)x+log|secx|+C`
C. `(1)/(4)tan^(4)x-(1)/(2)x+log|secx|+C`
D. none of these

Answer

C

Explanation

`I=inttan^(3)xtan^(2)xdx=inttan^(3)x(sec^(2)x-1)dx`.
`=inttan^(3)xsec^(2)xdx-inttan^(3)xdx=intt^(3)dt-inttanx(sec^(2)x-1)dx`, where tan x=t
`=(t^(4))/(4)-inttanxsec^(2)xdx+inttanxdx`
`=(1)/(4)tan^(4)x-intudu+log|secx|+C`, where tan x=u
`=(1)/(4)tan^(4)x-(1)/(2)u^(2)+log|secx|+C=(1)/(4)tan^(4)x-(1)/(2)tan^(2)x+log|secx|+C`.