We have: #int# #frac(sec^(2)(x))(1 - sin^(2)(x))# #dx#
#= int# #frac(sec^(2)(x))(cos^(2)(x))# #dx#
#= int# #frac(sec^(2)(x))(frac(1)(sec^(2)(x)))# #dx#
#= int# #sec^(2)(x) cdot sec^(2)(x)# #dx#
Then, the Pythagorean identity is #cos^(2)(x) + sin^(2)(x) = 1#.
We can divide through by #cos^(2)(x)# it to get:
#Rightarrow 1 + tan^(2)(x) = sec^(2)(x)#
Let's apply this rearranged identity to our integral:
#= int# #(1 + tan^(2)(x)) cdot sec^(2)(x)# #dx#
Now, let's use #u#-substitution, where #u = tan(x) Rightarrow du = sec^(2)(x)# #dx#:
#= int# #(1 + u^(2))# #du#
#= int# #1# #du + int# #u^(2)# #du#
#= u + frac(1)(3) u^(3) + C#
Finally, we can substitute #tan(x)# in place of #u#:
#= tan(x) + frac(1)(3) tan^(3)(x) + C#
