(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

do the following integrals converge?

i) [tex]\int_0^{1}\frac{dx}{x^{3/2}e^{x}}[/tex]

ii) [tex]\int_0^{1}\frac{x}{\sqrt{1-x^{2}}}dx[/tex]

3. The attempt at a solution

looking at them i can guess that they both diverge - proving this is the hard part - this is what i have got but it doesn't prove anything...

i) [tex]\frac{1}{x^{3/2}}[/tex][tex]\geq1[/tex]

[tex]\frac{1}{x^{3/2}e^{x}}[/tex][tex]\geq{e^{-x}}[/tex]

[tex]e^{-x}[/tex] converges

[tex]\frac{1}{e^{x}}[/tex][tex]\leq1[/tex]

[tex]\frac{1}{x^{3/2}e^{x}}[/tex][tex]\leq{x^{-3/2}[/tex]

[tex]x^{-3/2}[/tex] divereges

ii) [tex]x\leq1[/tex]

[tex]\frac{x}{\sqrt{1-x^{2}}}[/tex][tex]\leq{\frac{1}{\sqrt{1-x^{2}}}[/tex]

[tex]{\frac{1}{\sqrt{1-x^{2}}}[/tex] diverges (cannot prove it though)

[tex]{\frac{1}{\sqrt{1-x^{2}}}\geq1[/tex]

[tex]{\frac{x}{\sqrt{1-x^{2}}}[/tex][tex]\geq x[/tex]

x converges

any help would be much appreciated.

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# Improper Integrals

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