Simple Integral

  • Thread starter icystrike
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  • #1
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Homework Statement



[tex]\int sin(101x) sin^99(x) dx [/tex]

Homework Equations



Complex Number

The Attempt at a Solution



[tex] sin(101x) = \frac{e^{101ix}-e^{-101ix}}{2i} [/tex]
[tex] sin^99(x) = Im(e^{99ix}) [/tex]

Still trying...
 
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Answers and Replies

  • #2
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[tex]\sin x=\frac{e^{ix}-e^{-ix}}{2i}[/tex]
Does that help?

edit: Ah, sorry. Didn't see the mangled tex. just a minute.
 
  • #3
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There is an identity for [tex]sin^nx[/tex] which transforms it into a sum of regular sines. Perhaps that is a place to start.
 
  • #4
lurflurf
Homework Helper
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use reduction formulae
try an identity from elementary trigonometry such as
sin(101x)sin(9x)^9=[exp(101 i x)-exp(-101 i x)][exp(9 i x)-exp(-9 i x)]^9/2^10
from which (or otherwise) one may see that
sin(101x)sin(9x)^9=(1/512)(cos(20 x)-9 cos(38 x)+36 cos(56 x)-84 cos(74 x)+126 cos(92 x)-126 cos(110 x)+84 cos(128 x)-36 cos(146 x)+9 cos(164 x)-cos(182 x))
 

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