I am reading a research paper and am stuck at a point where the author uses angular integrals. I don't have any idea about it and would like help. The angular integral is:
$$I_k (y)=\int_0^\pi {\sin^2(\theta)\cos^{2k}(\theta) \over \beta^2(y)-\cos^2 (\theta)} d\theta$$ The author directly give results to this integral without proof. The results are: $$I_0 (y)=\pi \left(1-\sqrt{1-{1\over\beta^2 (y)} } \right),$$
$$I_1 (y)={-\pi \over 2}+\beta^2 (y)I_0 (y),$$
I don't know how he got these results. Can anyone please help me?
from Hot Weekly Questions - Mathematics Stack Exchange
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