Explore BrainMass
Share

Computing integrals via analytic continuation

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

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

© BrainMass Inc. brainmass.com October 10, 2019, 7:39 am ad1c9bdddf
https://brainmass.com/math/complex-analysis/computing-integrals-via-analytic-continuation-589748

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.

\$2.19