How do you integrate this problem involving trig functions and high exponents?

• Jan 18th 2009, 07:28 PM
Dana_Scully
How do you integrate this problem involving trig functions and high exponents?
I've tried this problem several times and end up with really crazy answers which are too embarrassing to state here. It's:

∫ cos(θ) sin(θ)^43 dθ

I tried with u = sinθ, which I'm not sure is right, and even then I can't integrate the resulting function.
• Jan 18th 2009, 07:31 PM
o_O
$\displaystyle \int {\color{blue}\cos (\theta)} \big[{\color{red}\sin (\theta)}\big]^{43} {\color{blue}\ d \theta}$

You're right! Using your sub we have: $\displaystyle {\color{red}u = \sin (\theta)} \ \Rightarrow \ {\color{blue}du = \cos (\theta) \ d\theta }$

Now simply substitute it all in. Can you go on from here?
• Jan 18th 2009, 07:37 PM
Dana_Scully
Thanks! Actually, I don't know where to go from:

∫(sin u)^(43) du

(Sorry I haven't learned the math environment yet.)
• Jan 18th 2009, 07:41 PM
o_O
No. $\displaystyle u$ is replaced with $\displaystyle \sin \theta$ NOT just $\displaystyle \theta$

$\displaystyle \int {\color{blue}\cos (\theta)} \big[\underbrace{{\color{red}\sin (\theta)}}_{\displaystyle = \ u}\big]^{43} {\color{blue}\ d \theta}$

So really you should have: $\displaystyle \int u^{43} du$
• Jan 18th 2009, 07:51 PM
Dana_Scully
Ah! Thank you so much. I have no idea how I could have mistaken that. So it's:

∫u^43 du

u^44 / 44

(sinθ)^44 / 44
• Jan 18th 2009, 08:21 PM
o_O
Don't forget the + C !