Let $$ \int_{0}^{ \infty} \frac{x+ \sin (x)}{1+x^2} dx, \ \int_{0}^{1} \frac{ \cos (x) }{ x^{ 3/2 } } dx, \ \int_{0}^{ \infty } \frac{ 1}{1+x} dx $$ be improper integrals we want to study.
I want to use comparison test to determine whether the following improper integrals are divergent or convergent. I found it easier to determine the convergent function, but have trouble dealing with divergent functions. Also, for normal integrals like second one, what is a good strategy to tell if it is convergent or divergent?
For (1) you only need to check in the interval $\;[1,\infty)\;$ (can you see why?) , and here
$$\frac{x+\sin x}{1+x^2}\ge\frac{x-1}{2x^2}$$
The same applies for the third case.
As for the second one on the given interval
$$\frac{\cos x}{x^{3/2}}\ge\frac{\cos1}{x^{3/2}}$$