Analyze the convergence of the following integrals using the comparison test

[Like Tony Stark]

Hi, as the title says, I am given the following integrals:
[MATH] 1) \int_0^1 \frac {cos^2 x}{(1-x^2)^{1/3} } \, dx [/MATH][MATH] 2) \int_0^1 \frac {ln(1+x^{1/3})}{e^{sinx}-1} \, dx [/MATH]

I have to use either direct comparison test or limit comparison test. For both of them I think that the limit comparison test is easier. Also, I used [MATH](1/x)^{2/3}[/MATH], which diverges.

But I have doubts when calculating the limit, because we have [MATH]sinx[/MATH] and [MATH]cosx[/MATH], and, as they oscillate, the limits don't approach to anything when [MATH]x[/MATH] tends to infinity.

 

[Deleted member 4993]

Hi, as the title says, I am given the following integrals:
[MATH] 1) \int_0^1 \frac {cos^2 x}{(1-x^2)^{1/3} } \, dx [/MATH][MATH] 2) \int_0^1 \frac {ln(1+x^{1/3})}{e^{sinx}-1} \, dx [/MATH]

I have to use either direct comparison test or limit comparison test. For both of them I think that the limit comparison test is easier. Also, I used [MATH](1/x)^{2/3}[/MATH], which diverges.

But I have doubts when calculating the limit, because we have [MATH]sinx[/MATH] and [MATH]cosx[/MATH], and, as they oscillate, the limits don't approach to anything when [MATH]x[/MATH] tends to infinity.

Please post one problem/thread - specially when those are very similar.

 

[Like Tony Stark]

Please post one problem/thread - specially when those are very similar.

I'll take it into account in future posts.