Explore BrainMass

Explore BrainMass

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

    Not what you're looking for? Search our solutions OR ask your own Custom question.

    This content was COPIED from BrainMass.com - View the original, and get the already-completed solution here!

    Evaluate using contour Integration

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

    © BrainMass Inc. brainmass.com March 5, 2021, 12:44 am ad1c9bdddf
    https://brainmass.com/math/integrals/integrate-from-zero-infinity-530969

    Solution Preview

    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 ...

    Solution Summary

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

    $2.49

    ADVERTISEMENT