# Improper integral comparison test

1. Jan 3, 2015

### hitemup

• Member warned about not using the homework template
The question asks whether the following converges or diverges.

$$\int_{0}^{\infty } \frac{\left | sinx \right |}{x^2} dx$$

Now I think there might be a trick with the domain of sine function but I couldn't make up my mind on this.
I tried to compare it with 1/x^2, (sinx)/x, and sinx. I actually expected that I would get something good with 1/x^2, but as the lower limit of the integral is zero, it ended up with infinity on (0, inf) and since 1/x^2 is greater than (sinx)/x^2, and is divergent as we just found, we cannot say whether the given function diverges or converges. So I'm wondering what is the right track on this question?

Last edited: Jan 3, 2015
2. Jan 3, 2015

### Stephen Tashi

$$\int_{0}^{\infty } \frac{\left | sinx \right |}{x^2} dx =$$

$$\int_{0}^{ \frac{\pi}{2} } \frac{\left | sinx \right |}{x^2} dx + \int_{ \frac{\pi}{2} }^{\infty} \frac{\left | sinx \right |}{x^2} dx$$

Deal with the two integrals separately.

In the interval $[0,\frac{\pi}{2}]$ you can establish an inequality between $|\sin(x)|$ and a non-periodic function. Try comparing it to $f(x) = x$.