Integral Help - integral of (cos^6)(x)

What is the integral cos^{6}(x)? I just want to make sure I am correct.

The answer I got was [(3sin(4x) + 2sin(2x) + 5x)/16] + C

but I have no clue if I am correct and am not very confident in my integration skills.

If your answer is different, it'd be a great help if you could show me the process you took to get there.

Thanks!

Re: Integral Help - integral of (cos^6)(x)

No that's not correct try using the reduction formula.

Re: Integral Help - integral of (cos^6)(x)

hint:

$\displaystyle \int cos^{m}(x)dx=\frac{sin(x)cos^{m-1}(x)}{m}+\frac{m-1}{m}\int cos^{m-2}(x)dx$

Re: Integral Help - integral of (cos^6)(x)

No I don't think it is correct because the derivative of [(3sin(4x) + 2sin(2x) + 5x)/16] + C

is not equal to cos6(x) . Moreover you didn't show to us how you did it...

Anyway use repeated integration by parts and you will get it...but be patient because the process is lengthy...

Good Luck

Re: Integral Help - integral of (cos^6)(x)

SMAD - I don't think the reduction formula is within the scope of my course, so if this question comes up in my exam I don't think I will get marks for process.

Do you know of another way to do it?

MINOANIMAN- Yeah, I thought so, My working took more than a page (possibly because of my relatively large handwriting) so I didn't post that sorry, but I broke up the cos^6(x) into (cos^2(x))^3 and substituted the half angle formula (cos^2(x) = 1/2(1+cos(2x)) and went from there, but I may have made some algebraic errors that I can't see... :\ I'm just wondering if there's a better way to break it up that will result in less possible errors.

Thanks for your help!

1 Attachment(s)

Re: Integral Help - integral of (cos^6)(x)

No you have to do as I said by repeated integration by parts without breaking into cos2x ....just follow the formula...I said it is lengthy

the correct answer is :

Attachment 29518