Is $\int_0^{\infty}\frac{x+3}{x^2+\cos x}$ convergent? 

You can use the limit comparison test. First note that for large $x$ 
\[\frac{x+3}{x^2 +\cos x} \approx \frac{x}{x^2}=\frac{1}{x}.\]
In other words
\[\lim _{x \rightarrow \infty} \frac{\frac{x+3}{x^2 +\cos x}}{\frac{1}{x}}=\lim_{x \rightarrow \infty} \frac{x^2+3x}{x^2 +\cos x}=\lim_{x \rightarrow \infty} \frac{x^2(1+\frac{3}{x})}{x^2(1+ \frac{\cos x}{x^2})} =\frac{1+0}{1+0}=1.\]
Thus by limit comparison test, the convergence of the given improper integral is the same as 
\[\int_0^{\infty}\frac{1}{x}dx,\]
which is divergent. Hence, $\int_0^{\infty}\frac{x+3}{x^2 +\cos x} dx$ also diverges.