The question is as follows, Determine for which $\alpha$ does the integral converge:

$$\int_3^\infty \frac{(x^{\alpha} +1)^{^\frac{2}{3}}}{(x+1)^{\frac{5}{3}}(x+2)} dx$$For the theory of testing the convergence of integrals, please view the Czech Math Tutor page (link)

We need to utilize a fact on the improper integral of polynomials

\begin{align*} \int_1^\infty \frac{1}{x^p} dx &= \begin{cases} \frac{1}{p-1} &p>1 \\ \infty &o.w. \end{cases} \end{align*}Here we need to make $\frac{8}{3} - \frac{2\alpha}{3} > 1$ to make our integral converge. Thus $\alpha < \frac{5}{2}$

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