Integral = I
I ( sec(2x))^3*tan(2x)dx = , 2x = u, 2dx = du, dx = du/2
1/2* I (secu)^3*tanudu = , secu = 1/cosu and tanu = sinu/cosu
1/2* I 1/(cosu)^3*sinu/cosudu =
1/2 I sinu/(cosu)^4* du = cosu=v, - sinudu =dv, and sinudu = - dv
1/2 I - dv/v^4 =
-1/2I v^(-4) dv =
-1/2 * v^(-4+1)/(-4+1) =
1/6*v^-3 = 1/(6v^3) = 1/(6(cosu)^3) =
1/6* 1/((cos(2x))^3 + C
OR
2x = u
dx = du/2
= 1/2*I sec^2u*secu*tanu du
secu =v and secu*tanu du=dv, sec^2u=v^2
!/2I v^2 dv =
1/6*v^3 = 1/6* sec^3u = 1/6*1/cos^3(2x) + C