How to integrate $\int_0^{\infty} \frac{dx}{1+x^3}$

Asked May 08 '21 at 21:10

Active May 08 '21 at 21:10

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How would one go about integrating such the function $$\int_0^{\infty} \frac{dx}{1+x^3}$$

The reason I am struggling with this is because normally i would use a contour ingrgration around a semi circle and then extend the radius up to infinity and minus infinity and then solve using cauchys integral theorem.

However, this function is odd and so i am wondering how one would go about integrating such a function since it would disappear over such limits.

asked May 08 '21 at 21:10

DJA

1

You could instead use partial fractions to factor the difference of cubed terms. – Тyma Gaidash May 08 '21 at 21:11
The question is from a worksheet concerning complex analysis, so would be intereseted to see if there is way that uses contour integrals that I just have not noticed. – DJA May 08 '21 at 21:13
@TymaGaidash would that work in this case? It might turn into an $\infty - \infty$ situation then. – RobertTheTutor May 08 '21 at 21:13
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@RobertTheTutor That is why we have improper integration with splitting the integral and using limits – Тyma Gaidash May 08 '21 at 21:14
It has been solved by the other question. Thanks for your input really apprecaiate it! – DJA May 08 '21 at 21:15
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@TymaGaidash Okay, I follow you now, thanks. Got a finite answer. – RobertTheTutor May 08 '21 at 21:36

How to integrate $\int_0^{\infty} \frac{dx}{1+x^3}$

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