# Trig Integrals

1. Jan 16, 2008

### Goldenwind

To find: Integral of "cos(x)^2 dx"
This becomes: Integral of "(u)^2 dx"
Where: u = cos(x), du/dx = -sin(x)

Problem is, du = -sin(x)dx... where do I get the -sin(x) from? If it was 16dx, I could multiply everything by 16/16, move 16/1 inside the integral, couple it with dx, and switch the 16dx for du... but am I allowed to multiply everything by -sin(x)/-sin(x)? The fact that it has x in it is what throws me off.

Showing the steps, how would you turn that dx into du?

Last edited: Jan 16, 2008
2. Jan 16, 2008

### neutrino

Yes, you are allowed to do so. After all, isn't sin(x) = sin(x) for all values of x?

But I would suggest that you employ Euler's formula to solve this particular problem. It's much simpler.

EDIT: The 'i' in the OP seems to have vanished! Anyway, either a simple trig identity involving the square of the cosine function or the Euler's formula can be used to solve the problem in a couple of steps.

Last edited: Jan 16, 2008
3. Jan 16, 2008

### Goldenwind

I wasn't familiar with this formula you speak of, so I did some quick searching, found this:
http://en.wikipedia.org/wiki/Eulerian_integral

Looked in my textbook... and it IS in there, but it's many many chapters ahead, and I don't understand that either. =/

Could you explain how I could apply it here? The format seems completely different, plus I'm doing a basic antiderivative, not finding the area from A to B :(

4. Jan 16, 2008

### neutrino

I was referring to this: http://en.wikipedia.org/wiki/Euler's_formula
http://mathworld.wolfram.com/EulerFormula.html

I suggested that because I saw an 'i' in your first post, and assumed that you were familiar with complex numbers.

See the edit. You must be familiar with trig. identities involving the the square of cos. (or you could look them up.) Use one of them.

5. Jan 16, 2008

### Goldenwind

Yeah, sorry about the i thing. Whenever I do homework on the computer, I use Sqrt() for square root, i() for integrals, etc. Just sorta like a personal system :)

6. Jan 16, 2008

### HallsofIvy

Staff Emeritus
No, you cannot multiply by sin(x) inside the integral and divide by sin(x) outside the integral!

Precisely because there is no "sin(x)" inside the integral already, you cannot use the "u= cos(x)" substitution. The standard way of integrating even powers of sine or cosine is to use the identities cos2(x)= (1/2)(1+ cos(2x)) and sin2(x)= (1/2)(1- cos(2x).

7. Jan 16, 2008