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    Computing integrals via analytic continuation

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    Using the identity:

    Integral from 0 to infinity of x^(-q)/(1+x) dx = pi/[sin(pi q)]

    valid for 0 < q < 1

    Compute the integral:

    Integral from 0 to infinity of x^p/(x^2+4 x + 3) dx

    for -1 < p < 1

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    Solution Preview

    The denominator of the integrand factors as follows:

    x^2+4 x + 3 = (x+1)(x+3)

    We can perform the partial fraction expansion:

    1/(x^2+4 x + 3 ) = 1/[(x+1)(x+3)] = 1/2 [1/(x+1) - 1/(x+3)]

    The integral can thus be written as

    I(p) = 1/2 Integral from 0 to infinity of x^p [1/(x+1) - 1/(x+3)] dx

    It now looks straightforward to split up the integrals into two ...

    Solution Summary

    We explain in detail how one can use analytic continuation of the given integral to compute the desired integral.