Take the integral:
integral sin(2 x) sec^2(x) dx
Simplify the integrand sin(2 x) sec^2(x) to get 2 tan(x):
= integral2 tan(x) dx
Factor out constants:
= 2 integral tan(x) dx
Rewrite tan(x) as (sin(x))/(cos(x)):
= 2 integral(sin(x))/(cos(x)) dx
For the integrand (sin(x))/(cos(x)), substitute u = cos(x) and du = -sin(x) dx:
= 2 integral-1/u du
Factor out constants:
= -2 integral1/u du
The integral of 1/u is log(u):
= -2 log(u) + constant
Substitute back for u = cos(x):
Answer: | = -2 log(cos(x)) + constant
