# Integration of (Cos X)^2

1. ### sony

104
Hi, I just don't get this. I'm to lazy to type inn what I've done, so I just took
a picture of my textbook:
http://home.no.net/erfr1/images/1.jpg
http://home.no.net/erfr1/images/2.jpg
You're supposed to end up with the starting point, right? So you can divide the rest by two?

Whatever I try, I get zero... I got the solution from mathematica, but don't understand a thing of it.

Thanks!

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3. ### Muzza

695
I'm not exactly clear on what it is you have done, but I'm guessing that you tried to integrate cos^2(x) using partial integration, and the equation you got reduced to 0 = 0? I suppose you expected to get back your original integral after a few iterations, so that you could solve for it. I wouldn't say that's what's "supposed" to happen (but it does happen, but not always, as you've demonstrated).

Instead of partial integration, use the identity cos(2x) = 2cos^2(x) - 1 <=> cos^2(x) = cos(2x)/2 + 1/2.

Last edited: Dec 28, 2004
4. ### sony

104
Yes, I tried partial integration. And thank you for the help!

Cheers

5. ### mathwonk

9,703
integration by parts does work of course but only if in the second step, you refrain from undoing the work of the first step. this can be confusing and is actually easier to do by guesswork.

i.e. the derivative of sincos is cos^2 - sin^2. but sin^2 + cos^2 = 1 is also easy to get as a derivative, namely it is the derivative of x. so the derivative of x + sincos is 2cos^2. now you are done after dividing by 2.

of course you notice here that cos^2 - sin^2 = cos(2x) is also coming in as in the trick suggested above, but here you do not have to know that trick.

6. ### dextercioby

12,314
Check post number 7 (mine :tongue2: ) from this thread:sine&cosine squared
I think it sould be pretty clear... U have both the primitives/antiderivatives and the definite integrals of the 2 functions wrt to the limits -\pi/2 and +\pi/2.

Daniel.

7. ### t!m

147
Make sure you read what Muzza said at the end. Just using the identity $$\text{cos}^2(x) = \frac{1}{2}(1+\text{cos}(2x))$$ makes this a very simple integral.

8. ### mathwonk

9,703
but the point is not everyone has this identity at their disposal.

9. ### TenaliRaman

646
Really!! I thought it was one of the basic identities in trigonometry usually referred to as the double angle formulae.

-- AI

10. ### itchy8me

21
hi,

i must integrate sin^2(x) by partial integration.

i I've done this by taking the 2nd partial ingeral and substituting it with the original integral of sin^2(x); and calculate 0 =0. what is it that i'm doing wrong?

what does mathwonk mean by : "integration by parts does work of course but only if in the second step, you refrain from undoing the work of the first step" ?

11. ### Geekchick

74
why must you use integration by parts? When it's very easy to integrate using the power reducing formula.

12. ### itchy8me

21
as far as i know, the power reduction formula is aquired by integration by parts. I must integrate by parts, because it can be done, and that is what the excercise says i must do: integrate using partial integration.

21
14. ### itchy8me

21
Doh!.. that's exactly the same method i was using, i still get 0=0 thus :(

Last edited: Mar 19, 2009
15. ### Count Iblis

So, it takes more than 4 years for Physics Forums to compute this integral? :uhh:

16. ### itchy8me

21
hahahah

i'm wondering how seriously i should take the "integrate by parts" bit, if i do a substitution in the second partial integral with a trigonometric identity, would that be considered cheating?

17. ### Geekchick

74
you can integrate by parts as long as you use the Pythagorean identity. I don't see how that would be cheating.

18. ### itchy8me

21
well my logic says that i might just as well then substitute sin^2(x) with an identity in the begining. So my question is now, is this integral solvable with integration by parts alone, or MUST i do a substitution somewhere along the line to beable to solve it?

19. ### Geekchick

74
Yes, never mind my earlier post I just made the process longer. You can absolutely do it by parts alone. Scratch that I messed up...Give me a minute

Last edited: Mar 19, 2009
20. ### Geekchick

74
Nope, I cannot see how it can be done without using any identities. As I said though your still using integration by parts.

21. ### itchy8me

21
so any hints in how i would do it with integration by parts alone?