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`int tan^3(pix/2)sec^2(pix/2) dx` Find the indefinite integral

`inttan^3(pix/2)sec^2(pix/2)dx`


apply integral substitution: `u=(pix)/2`


`=>du=(pi/2)dx`


`=>dx=(2/pi)du`


`inttan^3(pix/2)sec^2(pix/2)dx=inttan^3(u)sec^2(u)(2/pi)du`


Take the constant out,


`=(2/pi)inttan^3(u)sec^2(u)du`


Again apply integral substitution: `v=tan(u)`


`=>dv=sec^2(u)du`


`=2/piintv^3du`


Apply the power rule,


`=2/pi(v^(3+1)/(3+1))`


`=2/pi((v^4)/4)`


Substitute back `v=tan(u)`  and `u=(pix)/2`


`=1/(2pi)tan^4((pix)/2)`


Add a constant C to the solution,


`=1/(2pi)tan^4((pix)/2)+C`

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