Absolute convergence of $\int_{0}^{\infty} \frac{\sqrt{ x }\cos{(x)}}{x+100} \ dx $

Question

I have the integral $$\int_{0}^{\infty} \frac{\sqrt{ x }\cos{(x)}}{x+100} \ dx $$ of which I am interested in finding if it converges or not. I am limited to the Cauchy's criteria: $$\int^{+\infty}_{a} f(x) \ dx \ \text{converges}$$ if

$$\forall \epsilon > 0, \exists A_{0} > a\:\forall A',A''>A_{0}:\left\lvert\int_{A'}^{A''} f(x) \, dx\right\rvert < \epsilon$$

and elementary integration. Another answer of which it has $x+2013$ instead, but the answer is not so clear and I do not know how to proceed with the inner integral by elemenatary mean. How do I proceed with this problem?

Answer 1

HINT:

For $x>100$, we have

$$\left |\frac{\sqrt{x}\cos(x)}{x+100}\right|\ge \frac{|\cos(x)|}{2\sqrt{x}}$$

Now, break the integral into intervals for which the cosine is of one sign and develop a series that diverges and is dominated by the integral.

Can you proceed?