definite complex integrals ..help me plz

how ca I answer this integration by using matlab

 ∫(x^3+3)/((x^2+1)(x^2+4))    interval[-∞,∞] ?

I answer it by using calculation by using residue :

∴(z^2+1)(z^2+4)=(z-j)(z+j)+(z-j2)(z+j2)

z = +j

R_1=(-1+3)/((+j+j)(-1+4))=-1/3 j

z =j2

R_2=((-4)+3)/((-4+1)(j2+j2))=-1/12 j

I=-5/12 J(2πJ)=5/6 π

Answer 1

[Roger Stafford]

Your integral from minus infinity to plus infinity is divergent, because for large x the integrand behaves like 1/x, which would give log(x) as an integral, and log(x) becomes infinite. In other words, there is a non-zero residue at infinity and you are integrating through infinity, so the integral is not well-defined. It is as though you were integrating through a simple pole singularity on the finite complex plane, which would be ill-defined.

[Matthew Young]

Obviously it's too late now but the integral is convegent.

you can split it into \int x^3 / ((x^2 +1)(x^2+4)) + \int 3 / ((x^2 +1)(x^2+4)). The first term evaluates to 0 and the second term behaves lke 1/x^4 . You can split the second term up using partial fractions and then use arctan to evaluate the integral.

[Torsten]

This is only true if you mean the Cauchy Principal Value of the integral.

In the usual sense, the integral does not exist as @Roger Stafford noticed correcty.