# Convergence of an integral

1. Jun 9, 2006

### 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.

2. Jun 9, 2006

### George Jones

Staff Emeritus
Hint: on the required interval, sinx < x.

3. Jun 9, 2006

### shmoe

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

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?

4. Jun 11, 2006

### 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?