Question

How do you integrate sec^2(2x)(1+sin(2x)) dx?

`intsec^2(2x)(1+sin(2x))`

Answer 1

`I = (tan(2x) + sec(2x))/2 + C`

Explanation:

We start by letting `u = 2x`. Then `du = 2dx` and `dx = (du)/2`.

`I = 1/2int sec^2u(1 + sin(u))du`

`I = 1/2int 1/cos^2u(1 + sinu)du`

`I = 1/2int sec^2u + sinu/cos^2udu`

`I = 1/2int sec^2u + tanusecu du`

`I = 1/2int sec^2udu + 1/2int tanusecu du`

These are two known integrals.

`I = 1/2(tanu + secu) + C`

`I = (tan(2x) + sec(2x))/2 + C`

Hopefully this helps!