Question

How to integrate -cos^2x

Answer 1

Do you know about complex numbers already?

Answer 2

If not, you need to work with trigonometric identities rather then with complex numbers.

Answer 3

Can't remember

Answer 4

\[\Large \cos(2x)=\cos^2x-\sin^2x \] Any ideas how you want to continue from there?

Answer 5

No

Answer 6

you need to remember your trig identifies, what is sin^2(x) ?

Answer 7

It's cos raised to 2 and then x not cos2x

Answer 8

I know, and exactly there is the problem. See with all trigonometric problems you face during integration, you will see that you can't integrate them the way they're written. So you need to develop a way to rewrite them, or rearrange them.

Answer 9

Yes

Answer 10

Alright, so consider the following equation:
\[\Large \cos(2x)=\cos^2x-(1-\cos^2x)=2\cos^2x-1 \]

Answer 11

Ok

Answer 12

Now from there it should be pretty obvious what to do next,

Answer 13

\[\Large 2\cos^2x-1=\cos(2x) \]

Answer 14

I'm blanking

Answer 15

Solve for cos^2(x) and substitute that back into your integral.

\[\Large \cos^2(x)=\frac{1}{2}\left( \cos(2x)+1\right) \]

Answer 16

See what you have achieved by that is the following, you express the quadratic expression of a trigonometric function in form of double angular formulas of the equal trig identities, the LHS you can't integrate, but the integral becomes pretty easy for the RHS.

Answer 17

Oh ok. I was tryin to do it using the reduction formula