SOLUTION: Find the exact value of the integral {{{int((sinx)^m/((sinx)^m + (cosx)^m), dx, 0, pi/2)}}} for all values of m.

Algebra ->  Trigonometry-basics -> SOLUTION: Find the exact value of the integral {{{int((sinx)^m/((sinx)^m + (cosx)^m), dx, 0, pi/2)}}} for all values of m.      Log On


   

Question 453405: Find the exact value of the integral int%28%28sinx%29%5Em%2F%28%28sinx%29%5Em+%2B+%28cosx%29%5Em%29%2C+dx%2C+0%2C+pi%2F2%29 for all values of m.
Answer by robertb(5830) About Me  (Show Source):
You can put this solution on YOUR website!
Let x = pi%2F2+-+y. Substitute for x in the integral.
int%28%28sinx%29%5Em%2F%28%28sinx%29%5Em+%2B+%28cosx%29%5Em%29%2C+dx%2C+0%2C+pi%2F2%29
=
=
= +int%28%28cosy%29%5Em%2F%28%28cosy%29%5Em+%2B+%28siny%29%5Em%29%2Cdy%2C+0%2Cpi%2F2%29%0D%0A = +int%28%28cosx%29%5Em%2F%28%28cosx%29%5Em+%2B+%28sinx%29%5Em%29%2Cdx%2C+0%2Cpi%2F2%29%0D%0A
==> int%281%2C+dx%2C+0%2C+pi%2F2%29 - +int%28%28cosx%29%5Em%2F%28%28cosx%29%5Em+%2B+%28sinx%29%5Em%29%2Cdx%2C+0%2Cpi%2F2%29%0D%0A = int%28%28sinx%29%5Em%2F%28%28sinx%29%5Em+%2B+%28cosx%29%5Em%29%2C+dx%2C+0%2C+pi%2F2%29
<==> int%281%2C+dx%2C+0%2C+pi%2F2%29 = 2%2Aint%28%28sinx%29%5Em%2F%28%28sinx%29%5Em+%2B+%28cosx%29%5Em%29%2C+dx%2C+0%2C+pi%2F2%29
==> %281%2F2%29%2A%28pi%2F2%29 = int%28%28sinx%29%5Em%2F%28%28sinx%29%5Em+%2B+%28cosx%29%5Em%29%2C+dx%2C+0%2C+pi%2F2%29
Therefore