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 March 5, 2021, 1:26 am ad1c9bdddf
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 ...
We explain in detail how one can use analytic continuation of the given integral to compute the desired integral.