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

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January 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.
January 18th 2009, 07:31 PM
o_O

$\int {\color{blue}\cos (\theta)} \big[{\color{red}\sin (\theta)}\big]^{43} {\color{blue}\ d \theta}$

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

Now simply substitute it all in. Can you go on from here?
January 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.)
January 18th 2009, 07:41 PM
o_O

No. $u$ is replaced with $\sin \theta$ NOT just $\theta$

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

So really you should have: $\int u^{43} du$
January 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
January 18th 2009, 08:21 PM
o_O

Don't forget the + C !

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