Integrate sec^22x

To integrate sec^22x, also written as ∫sec²2x dx, sec squared 2x, (sec2x)^2, and sec^2(2x), we start with a u substitution.

Let u = 2x. This is a simple u substitution.

Therefore du/dx = 2. This is a simple differentiation step.

We rearrange the previous expression for dx.

We now have another integration on the RHS that means the same thing but is in terms of u.

We move the constant 1/2 outside of the integral to simplify.

We recall a standard integration solution usually found in formula booklets. We also proved it in the article integrate sec^2x.

Therefore in our case, we can integrate in terms of u, to give the answer, as shown above.

We substitute the answer back into equation [1] and reintroduce the constant 1/2.

We recall that u=2x, and we can substitute that back in as well.

Hence this is the final answer.