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I have learned how to solve equations like:

[tex]\int^{2\pi}_{0}\frac{1}{1+sin\theta}d\theta[/tex]

using

[tex]\int^{2\pi}_{0}F(sin\theta,cos\theta) d\theta[/tex]

[tex]\int_{C}F(\frac{z-z^{-1}}{2i},\frac{z+z^{-1}}{2}) \frac{dz}{iz}[/tex]

How do I solve equations of the type:

[tex]\int^{\pi}_{0}\frac{1}{1+sin\theta}d\theta[/tex]

and

[tex]\int^{\pi}_{-\pi}\frac{1}{1+sin\theta}d\theta[/tex]

Please help :-)