Question

What is the integral of `cos2(theta)`?

Answer 1

`int cos(2theta) "d"theta = 1/2 sin(2theta) + C`,

where `C` is an integration constant.

Explanation:

I think you mean `cos(2theta)` instead of `cos2(theta)`.

If you know that `int cos(x) dx = sin(x) + C`, then we can use a substitution (which is the reverse of the chain rule).

Let `u = 2theta`,

`frac{"d"u}{"d"theta} = 2`.

So,

`int cos(2theta) "d"theta = 1/2 int cos(2theta) * (2) "d"theta`

`= 1/2 int cos(2theta) * frac{"d"u}{"d"theta} "d"theta`

`= 1/2 int cos(u) "d"u`

`= 1/2 (sin(u) + C_1)`,
where `C_1` is an integration constant.

`= 1/2 sin(2theta) + C_2`,
where `C_2 = 1/2 C_1`.