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When I try to solve the integration f(x) as following (type it in the Mathematica) $ f(x)= \frac{1}{π} \int_{-1}^{1} \frac{\sqrt{1-y^2}}{(y-x)}dy $, I met some boundary problems. I have searched some basic command like NIntegrate or PrincipalValue, but it dose not work. I think these commands just for finding a value of integration, but not for calculating a function. Can anyone provide any suggestion?

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Though M recognizes lot of types of integration patterns but still there are some that it can calculate only numerical value of using NIntegrate. – Rorschach Aug 23 '13 at 20:41
I do not understand the question. Mathematica is giving the conditions for finding the result of the integration. If you use these, you'll get the result. 1/Pi Assuming[Element[x, Reals] && x >= 1 , Integrate[ Sqrt[1 - y^2]/(y - x), {y, -1, 1}]] gives -x + Sqrt[-1 + x^2] !Mathematica graphics – Nasser Aug 23 '13 at 20:42
I am so sorry and I forgot to say that -1<x<1. – Henry Aug 23 '13 at 21:29
I am not sure it can be integrated in Mathematica and I am trying to do analytic calculation. – Henry Aug 23 '13 at 21:30
but if -1<x<1 then there is a pole when x=y. It does not matter if it is analytical or numerical. A pole is a pole is a pole. I am confused, but may be I am missing the point still. Numerical integration can sometimes handle limited number of poles, if you are lucky and so on. But for analytical integration, I do not see how to avoid this. – Nasser Aug 23 '13 at 21:49

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