You're trying to simply copy/paste what I posted on the other site, and it didn't work very well because my post contained LaTeX. This is what I posted:
According to W|A the definite integral given in #4 does not converge.
[MATH]I=\int_2^{\infty}\frac{x\sqrt{x}}{x^2-1}\,dx[/MATH]
Using your substitution:
[MATH]u=\sqrt{x}\implies du=\frac{1}{2\sqrt{x}}\,dx\implies dx=2u\,du[/MATH]
We then have:
[MATH]I=2\int_{\sqrt{2}}^{\infty}\frac{u^4}{u^4-1}\,du=2\int_{\sqrt{2}}^{\infty}1+\frac{1}{u^4-1}\,du[/MATH]
And then using partial fractions, we may write:
[MATH]I=\frac{1}{2}\int_{\sqrt{2}}^{\infty} 4+\frac{1}{u-1}-\frac{1}{u+1}-\frac{2}{u^2+1}\,du[/MATH]
Do you agree so far?
And I have asked you to neatly show how you'd proceed, because your hot-linked images are hard to read.
