by HallsofIvy » Wed Mar 06, 2019 12:14 pm
First, change the order of integration:
[tex]A^2B^2C^2\int_{y= 0}^{2\pi}\int_{x= 0}^\pi \frac{sin(x)}{A^2B^2 cos^2(x)+ A^2C^2 sin^2(x)sin^2(y)+ B^2C^2 sin^2(x)cos^2(y)} dxdy[/tex].
Now change each [tex]sin^2(x)[/tex] to [tex]1- cos^2(x)[/tex]:
[tex]A^2B^2C^2\int_{y= 0}^{2\pi}\int_{x= 0}^\pi \frac{sin(x)}{A^2B^2 cos^2(x)+ A^2C^2 (1- cos^2(x))sin^2(y)+ B^2C^2 (1- cos^2(x))cos^2(y)} dxdy[/tex].
- Make the substitution u= cos(x) so that [tex]du= - sin(x)dx[/tex] and the integral becomes
[tex]A^2B^2C^2\int_{y= 0}^{2\pi}\int_{x= 0}^\pi \frac{-1}{A^2B^2 u^2+ A^2C^2 (1- u^2)sin^2(y)+ B^2C^2 (1- u^2)cos^2(y)} dudy[/tex]
See if you can do anything with that.