I am looking to figure out if the following integral converges or diverges. $\int_{0}^\infty{\frac{dx}{\sqrt{x}*(1+x)}}$ = $\int_{0}^\infty{\frac{dx}{\mathrm{x}^{1/2}+\mathrm{x}^{3/2}}}$.
I have set it less than or equal to $\int_{0}^\infty{\frac{dx}{\mathrm{x}^{3/2}}}$. Since there is an asymptote at $x = 0$, I split this integral into $\int_{0}^1{\frac{dx}{\mathrm{x}^{3/2}}}$ + $\int_{1}^\infty{\frac{dx}{\mathrm{x}^{3/2}}}$.
However this results in a contradiction because the p test says the first part should diverge and the second part of this split integral should converge. Therefore, I cannot say anything about the convergence of the original integral. I'm not sure what to do at this point.
The key to a split like this? Split the integral before you make a comparison, then compare to different things in the two pieces. If you were going to make the same comparison in both parts, there wouldn't be any reason to split it. (OK, that's a more advanced view - with an improper Riemann integral, we have to split anyway)
For $1\le x<\infty$, $x^{\frac12}\le x^{\frac32}$ and the denominator is dominated by the $x^{\frac32}$ term, so we compare to $x^{-\frac32}$. For $0<x\le 1$, $x^{\frac12}\ge x^{\frac32}$ and the denominator is dominated by the $x^{\frac12}$ term, so we compare to $x^{-\frac12}$ instead.