# Integral from 0 to infinity of x^3 sin(2x)/(x^2+1)^2

I need the solution of:

integral from 0 to infinity of x^3 sin(2x)/(x^2+1)^2 dx

https://brainmass.com/math/integrals/residue-theorem-442849

#### Solution Preview

Put

I = integral from 0 to infinity of x^3 sin(2x)/[(x^2+1)^2] dx

Then because the integer is an even function, the integral from minus infinity to infinity is twice as large:

2 I = integral from -infinity to infinity of x^3 sin(2x)/[(x^2+1)^2] dx

We can replace sin(2x) by the imaginary part of exp(2 i x) in here:

2 I = integral from -infinity to infinity of x^3 Im[exp(2 i x)]/[(x^2+1)^2] dx

= Im [ integral from -infinity to infinity of x^3 exp(2 i x)/[(x^2+1)^2] dx ]

With the replacement of sin(2x) by Im[exp(2 i x)], we can now complete the contour by adding a half circle in the upper half plane, the integral along this circle will vanish in the limit of the radius going to infinity. Note that in some cases you can have a factor x in ...

#### Solution Summary

We explain how to compute this integral using contour integration methods.