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Evaluate $\int_{-1}^{1} \frac{\sqrt{1-x^2}}{1+x^2}dx$ using the residues.

I am not really sure what contour I should be considering.

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Hint: Rewrite your integral in terms of trig functions. Your integral is equal to $$ \int_{-\pi/2}^{\pi/2}\frac{\cos^2 t}{1+\sin^2 t}dt $$

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    and then am guessing I should use the exponential formulae for both sin and cos. Thanks :) – 2017-01-13
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    @debanjana sounds right! – 2017-01-13
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    I think we try to compute $I+\pi = I+ \int_{-\pi/2}^{\pi/2} dt = \int_{0}^{2\pi} \frac{1}{1+\sin^2 t} dt$ and then apply the formula for exponential it becomes easier! – 2017-01-14