Calculus of a Single Variable, Chapter 8, 8.3, Section 8.3, Problem 24

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