Integrate: Power of cos is even and non-negative

1. Sep 2, 2007

Chele

First off, thanks for your help. I feel like a dope because everything I'm doing looks correct to me, but my answer is not the same as the book or solution manual gives! I'm usually so good at finding my mistakes.

Thanks for taking the time to help.

1. The problem statement, all variables and given/known data

2. Relevant equations

Where did I go wrong??

3. The attempt at a solution

The book's answer is -1/4 (cos x)^4 + C even when I try to convert sines to cosines, I'm not getting that those two answers were the same.

2. Sep 2, 2007

bel

After the second step, try substituting $$u = cos (x)$$ instead. By the way, the two are equal, try substituting $$sin^2 (x) = (1-cos^2 (x))$$ for every $$sin^2 (x)$$ in the answer and remember that $$\frac{1}{4}$$ is a constant as well.

Last edited: Sep 2, 2007
3. Sep 2, 2007

Chele

Thanks...I was able to get the book's answer, however...what is wrong with my solution?

4. Sep 2, 2007

rock.freak667

Nothing looks wrong with your solution...i think your book just wanted you to immediately realize that u=cos(x) was a good substitution when you saw the integrand

5. Sep 2, 2007

Chele

I have to go to work, I'll try the substitution again. Maybe just bad algebra trying to substitute. I'll post later if I got it or not.

Thanks.

6. Sep 2, 2007

durt

Substitute $$\sin^2{x} = 1 - \cos^2{x}$$ into your answer, and you should get the book's. Hence what you did wasn't wrong, just inefficient.

7. Sep 3, 2007

Chele

Duh...got it....I think.

Okay, thanks. It took a moment for it to sink in. When I substituted to try to get the two answers to equal up, I got $$\frac{1}{4} -$$ $$\frac{1}{4} cos^{4}x$$

I see now that the extra $$\frac{1}{4}$$ is just part of the constant of integration.

If that is a correct statement, I am straight.

Thank you!

8. Sep 3, 2007

HallsofIvy

Staff Emeritus
By the way, you kind of confused me with your title: "Power of cos is even and non-negative". Obviously, here, both sine and cosine are to odd powers. The cosine substitution is the "standard" one for a situation like this.

9. Sep 3, 2007

Chele

Thanks for the help..... It's all new to me!!!

Last edited: Sep 3, 2007
10. Sep 3, 2007

HallsofIvy

Staff Emeritus
Any time you have an integrand of the form sinn(x)cosm(x) with either m or n odd, you can always factor out 1 of those (for example if sin5(x)), so that you have left an even power (sin(x) sin4(x)). Use cos2(x)= 1- sin2(x) or sin2(x)= 1- cos2(x), repeatedly if necessary, to write it in terms of the other function. (sin4(x)= (sin2(x))2= (1- cos2(x))2 and then use the substitution u= cos(x) or u= sin(x).

If both m and n are even then it is harder. Use sin2(x)= (1- 2sin(2x))/2 and cos2(x)= (1+ 2cos(2x))/2 to reduce the powers until you have an odd power of one or the other.

11. Sep 3, 2007

Chele

Umm...you know what...you are absolutely right. Cos and sin are odd and positive. I work midnight shift and I posted it after not having sleep for quite some time.

As for the even powers, I'm moving on to that homework tonight.

Again, appreciate the help. This place is a gold mine of information!