# Trigonometric Integral

1. Oct 19, 2009

### Mugen Prospec

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

$$\int$$ 41(sin$$^{6}$$(x))(cos$$^{3}$$(x))

2. Relevant equations

3. The attempt at a solution

I think you are supposed to use the half angle identities and then maybe integration by parts but Im lost on it.

2. Oct 19, 2009

### lanedance

how about starting by trying the substitution u = sinx

3. Oct 19, 2009

### Mugen Prospec

I have tried a few different things, more than I wanted to list but it just keeps getting more and more convoluted. I need a walk through we just started this in class and my teacher doesnt answer questions so Im just a bit lost over all.

4. Oct 19, 2009

### Dick

You don't have to do anything complicated with it, cos(x)dx=d(sin(x)). Just substitute u=sin(x). There are rules for dealing with powers of sin's and cos's. They are particularly easy if one power is odd.

5. Oct 19, 2009

### Mugen Prospec

Ok I know what your talking about. It was just in our chapter about integration by parts so I was a little first sight shocked. Can some one give me an answer so when I complete it I can know if I am correct or not.

6. Oct 19, 2009

### Dick

Oh, come on. Just work it out and show us what you get. I'll guarantee someone will check it.

7. Oct 19, 2009

### Mugen Prospec

Ok I have other work to do this one has bee on my mind all night. Ill post it tomorrow when I am clear of thought

8. Oct 19, 2009

### Dick

It's REALLY easy with the substitution lanedance suggested. You might want to clear your mind on this one and go to bed happy. But tomorrow is ok too.

9. Oct 19, 2009

### Mugen Prospec

Ok thanks alot Ill see what I can do tonight.

10. Oct 19, 2009

### Dick

Free hint since you are playing along: cos^2(x)=1-sin^2(x)=(1-u^2).

11. Oct 19, 2009

### Mugen Prospec

ok i got
41(sin^7(x)/7)(1/2 x + 1/4sin2x+c)

12. Oct 20, 2009

### lanedance

doesn;t look quite right to me, maybe show your working

13. Oct 20, 2009

### Mugen Prospec

Thats what I was thinking. after using u substitution I was left with

(u)^6 (cos^2(x)) cos(x) du/cos(x)

So cos(x) canceled out.

Last edited: Oct 20, 2009
14. Oct 20, 2009

### Dick

Fine. Now what's cos^2(x) in terms of u?

15. Oct 20, 2009

### Mugen Prospec

Oh ok the identity. so now all turns into
u^7 (1-u^2) du
Do I do their antiderivative now? And then substitute sin(x) back in. Im not sure if you can do each of there AD since they are multiplying one another.

16. Oct 20, 2009

### Dick

Of course you don't take the AD of each one. That's wrong. You multiply it out.

17. Oct 20, 2009

### Mugen Prospec

OK that what I thought Im trying to do this not with out witting it down since Im in chemistry.
So we get (u^6)-(u^8)du
then (u^7)/7) - (u^9)/9)
41 (sin^7(x))/7) - (sin^9(x))/9)
is that it maybe?

18. Oct 20, 2009

### Dick

You are missing a parenthesis level following the 41, but yes, that's it.

19. Oct 20, 2009

### Mugen Prospec

Ok awesome thank you for you patience.