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Integrate 1/(1 + x^a) from zero to infinity

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Evaluate using contour Integration

J(a) = integral from zero to infinity of (dx/(1 + x^a)) ; a> 1

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Solution Summary

I show how the integral can be computed using the methods of complex analysis.

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We can rewrite the integral using the substitution x^a = u as:

J(a) = 1/a Integral from 0 to infinity of u^(1/a -1)/(1+u) du

Putting p = 1/a - 1, we thus have to evaluate:

Integral from 0 to infinity of x^p/(1+x) dx

for -1 < p < 0

To compute the integral using contour integration, we define the function z^p in the complex plane as follows. For any given z we define polar coordinates r and theta, such that z = r exp(i theta) and we choose theta such that 0 < theta < 2 pi. We then define z^p = r^p exp(i p theta). We thus put the branch cut along the positive real axis. Then we consider the contour integral of z^p/(1+z) along the following contour. We start at the point epsilon + i delta and move parallel to the real axis to i delta + R (we need to stay clear of the ...

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