anonymous
  • anonymous
how do you get a formula for an integral from 0 to pi of sine of theta raised to the power of 2n where n is an integer from 0?
Mathematics
  • Stacey Warren - Expert brainly.com
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SOLVED
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jamiebookeater
  • jamiebookeater
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anonymous
  • anonymous
\[\int\limits_{0}^{\Pi}\sin ^{2n}(\theta)d \theta\]
anonymous
  • anonymous
2n+1 is the exponent and the denomater is 2n+1
anonymous
  • anonymous
actually i think thats wrong sry ill look it up

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anonymous
  • anonymous
The answer is \[\Pi(2n)!/((2^{2n})\times (n!)^{2})\]
anonymous
  • anonymous
but how do you get to that?
anonymous
  • anonymous
So I know that e^(i*pi/2)= i
anonymous
  • anonymous
and e^(i(theta)) = cos(theta)+ isin(theta)
anonymous
  • anonymous
also cos(theta)= (e^(itheta)+e^(-itheta))/2 and sin(theta)=(e^(itheta)-e^(-itheta))/(2i)
anonymous
  • anonymous
There is also an identity, 2cos(theta)e^(itheta)=1+e^(i2theta)
anonymous
  • anonymous
You there?
anonymous
  • anonymous
yea chill im just trying to figure it out
anonymous
  • anonymous
Moo is using De Moivre's Theorem
anonymous
  • anonymous
Exactly!
anonymous
  • anonymous
Then.....?
anonymous
  • anonymous
also not to mention, \[\int\limits_{0}^{\Pi}\cos \theta ^{n}\cos n \theta d \theta=\Pi/2^{n}, n =0\]
anonymous
  • anonymous
n=0, 1, 2, 3....
anonymous
  • anonymous
this might also be a hint, but e^(i*2pi*k) where k is any integer is 1. (for reasons I fully don't know)
anonymous
  • anonymous
Whats your major Moo?
anonymous
  • anonymous
english
anonymous
  • anonymous
contemporary american drama
anonymous
  • anonymous
you know an awful lot about calculus for an English major..... lol
anonymous
  • anonymous
read Oleanna, Third, ah...Shape of things and such. You know calculus is really fun
anonymous
  • anonymous
so rewrite the integrand as sin(theta)*sin(theta) both raised to the power n?
anonymous
  • anonymous
calculus is legit
anonymous
  • anonymous
also is cos(n*theta)=cos(theta)^n? Yeah, it is..lol.
anonymous
  • anonymous
oh, bedtime. my mom is calling me downstairs with her awful hammer thingie.
anonymous
  • anonymous
Thanks a lot guys.

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