definite complex integrals ..help me plz
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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
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Answers (1)
Roger Stafford
on 14 Mar 2015
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.
2 Comments
Matthew Young
on 18 Jan 2022
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
on 18 Jan 2022
This is only true if you mean the Cauchy Principal Value of the integral.
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