How can one know that a given integral diverges?

Question

In a d-dimensional integral of a rational function,How can we know that an integral diverges or converges depending on d and the highest powers of the numerator and denominators? What I understand is that if a 4-dimensional minkwoski integral is spherically symmetric with respect to the origin ,One can transform $d^{3}k$ to spherical coordinates and so obtain a function of one variable which is easy but I don't know how to generalize this to the general d-dimensional minkwoski spacetime ?

Answer 1

$$\int^\Lambda d^dk \frac{k^n}{(k^{2}-\mu^2)^m}\approx\int^\Lambda dk k^{d-1} \frac{k^n}{(k^{2}-\mu^2)^m}\approx \Lambda\Lambda^{d-1+n-2m}=\Lambda^{d+n-2m}$$

this is the behaviour of the integral in the ultraviolet regime, namely $k$ that goes to infinity. You have to think $\Lambda$ as a constant (cut-off) and then push it to infinity to understand the kind of divergence. Remind that: $$\int^\Lambda dk \frac{1}{k}=\log\Lambda \quad log\quad divergent$$

$$\int^\Lambda dk =\Lambda \quad linearly\quad divergent$$

$$\int^\Lambda dk k =\Lambda^2 \quad quadratically\quad divergent$$