converge/diverge: int[1,infty][cbrt[x^2+2] sqrt[x^3+3]]dx

[cheffy]

This is the last one, I think/hope!

\(\displaystyle \[ \int_1^\infty \frac{\sqrt[3]{x^2+2}} {\sqrt{x^3+3}} \,dx.\]\)

Converge or diverge?

I have no idea how to simplify this. =\

 

[tkhunny]

Is it any easier like this? \(\displaystyle \sqrt[6]{\frac{(x^{2}+2)^{2}}{{(x^{3}+3)^{3}}}\)?

 

[cheffy]

umm...not really. >_<

 

[tkhunny]

Okay, then first guess from the way it is written...

\(\displaystyle \frac{x^{2/3}}{x^{3/2}}\;=\;x^{-5/6}\)

Since -1 < -5/6 < 1, I suspect it diverges. Now prove it.

 

[cheffy]

Would I multiply out (x+2)^2 and then take the largest power and do the same as the bottom? And then compare that to the original?