Why can't a value for this definite integral be found? I was trying to find out if  $\int _0^{\infty }\:\frac{1}{\left(1+x^3\right)^{\frac{1}{2}}}dx$ converges or diverges. I split it into a sum, that is
$\int _0^{1 }\:\frac{1}{\left(1+x^3\right)^{\frac{1}{2}}}dx$ + $\int _1^{y }\:\frac{1}{\left(1+x^3\right)^{\frac{1}{2}}}dx$
where $y>1$ . I found that the second integral of the sum converges as y tends to infinity, but I can not evaluate the first integral. When I plot the graph, I can see clearly that the area under the curve of $f(x)=\frac{1}{\left(1+x^3\right)^{\frac{1}{2}}}$ with x in $[0,1]$ is finite. What is wrong with that integral?

Let $u^2 = 1+x^3 \to 2udu = 3x^2dx = 3(u^2-1)^{2/3}dx \to I = \displaystyle \int_{1}^{\sqrt{2}} \dfrac{2}{3\left(u^2-1\right)^{2/3}}du$. Next you can make a substitution: $u = \sec \theta$.
