Please see the attached file.

First, the p-integral test:

The integral is convergent if and only if p>1

Then, the comparison test:

If f(x) and g(x) are functions defined on [a,b] such that for any then we have:

If is convergent, so is
If is divergent, so is

For the first integral we note that at infinity the integrand approaches an asymptotic behavior of 1/x:

Thus we can approximate:

This is a p-integral with p=1, and this integral diverges.

Therefore diverges as well.

We start with a simple substitution:

We can invert the limits and the sign:

The only problematic point here is u=0, however employing L'hopital rule we see that the integrand has a well defined value at u=0

Thus, the integrand is well defined over the entire finite interval; hence this integral has a finite value.

Therefore, the integral converges absolutely.

The integral is known as the Fresnel cosine integral.

Integrating by parts we have:

The term is finite:

As for the integral we can show that it converges absolutely using the comparison and p-integral tests together:

Hence:

But is a p-integral with p>1, therefore the integral converges, and as a direct consequence so is the integral

Thus the integral converges absolutely, and therefore

Converges absolutely.