# Computing integrals via analytic continuation

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.

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