Mathematics

Problem

Consider the improper integral

I=\int_{1}^{\infty} \frac{x}{x^{3}+1}\,dx.

The integral is presented without any request to compute its numerical value. (a) State the interval(s) on which the integral is improper and identify the corresponding type(s) of improper behavior. (b) Determine whether

I

converges or diverges. Provide a complete justification that includes an explicit comparison with an elementary function defined on

[1,\infty)

. (c) Suppose the numerator of the integrand is replaced by

x^{r}

, where

r

is a real parameter. Without evaluating any integral, determine all

r

for which

\int_{1}^{\infty} \frac{x^{r}}{x^{3}+1}\,dx

converges. Support your conclusion with appropriate reasoning.

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