Is this Integral Convergent or Divergent?

[Metal]

How do I know whether this is convergent or divergent:

Integral of (x³+1)/((sinx)^1/2) dx between 0 and pi/2

I know that this integral is convergent if Lim n->0 of Integral of (x³+1)/((sinx)^1/2)) dx between n and pi/2 exists and is not infinite (why is that?). Otherwise its divergent.

So I thought I should find the antiderivative F of (x³+1)/((sinx)^1/2)) and then calculate F(pi/2) - F(n), the problem being that i don't know how to find this F, and I don't think that this is what I'm supposed to do.

Appreciate any help.

 

[George Jones]

Hint: on the required interval, sinx < x.

 

[shmoe]

I know that this integral is convergent if Lim n->0 of Integral of (x³+1)/((sinx)^1/2)) dx between n and pi/2 exists and is not infinite (why is that?). Otherwise its divergent.

Where is this function continuous? That will answer why you only have to worry about the left endpoint.

So I thought I should find the antiderivative F of (x³+1)/((sinx)^1/2)) and then calculate F(pi/2) - F(n), the problem being that i don't know how to find this F, and I don't think that this is what I'm supposed to do.

Try comparing it with an integral you know converges. The sin(x) is the troubling part, but you're near 0 so can you think of something nicer to compare it with?

 

[Metal]

f(x) = (x³+1)/((sinx)^1/2)) ~ g(x) = 1/(x^1/2) near 0 because sinx/x = 1 near 0 (and x³+1/1 too).
Since 1/(x^1/2) is convergent near 0 then f(x) also is.

Is that right?