Here is your solution step-by-step in non-LaTex everyday language!.
Take the integral:
integral sec^3(x) dx
Use the reduction formula, integral sec^m(x) dx = (sin(x) sec^(m-1)(x))/(m-1) + (m-2)/(m-1) integral sec^(-2+m)(x) dx, where m = 3:
= 1/2 tan(x) sec(x)+1/2 integral sec(x) dx
Multiply numerator and denominator of sec(x) by tan(x)+sec(x):
= 1/2 tan(x) sec(x)+1/2 integral(sec^2(x)+tan(x) sec(x))/(tan(x)+sec(x)) dx
For the integrand (sec^2(x)+tan(x) sec(x))/(tan(x)+sec(x)), substitute u = tan(x)+sec(x) and du = (sec^2(x)+tan(x) sec(x)) dx:
= 1/2 tan(x) sec(x)+1/2 integral1/u du
The integral of 1/u is log(u):
= (log(u))/2+1/2 tan(x) sec(x)+constant
Substitute back for u = tan(x)+sec(x):
= 1/2 tan(x) sec(x)+1/2 log(tan(x)+sec(x))+constant
Factor the answer a different way:
= 1/2 (tan(x) sec(x)+log(tan(x)+sec(x)))+constant
Which is equivalent for restricted x values to:
Answer: |= 1/2 (tan(x) sec(x)-log(cos(x/2)-sin(x/2))+log(sin(x/2)+cos(x/2)))+constant