Solving Integral of "(u)^2 dx": Steps to Turn dx into du

[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?

 

[neutrino]

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.

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.

 

[Goldenwind]

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.

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 :(

 

[neutrino]

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. =/

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.

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 :(

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.

 

[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 sort of like a personal system :)
Thanks for your help!

 

[HallsofIvy]

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).

 

[rocomath]

You need to know your tirg identities by heart.

http://www.mathlinks.ro/Forum/weblog_entry.php?p=968378#968378