I missed up the last lecture and can't understand how to determine whether an integral with parameters is convergent or divergent?
For example: For which values of the parameters $p,q \in [0,\infty)$, the following integral is convergent
$$
\int_{0}^{+\infty} \frac{dx}{x^p + x^q}.
$$
Any help would be greatly appreciated.
Best Answer
Without loss of generality we suppose $p\geq q$ then $$\frac{1}{x^p+x^q}=\frac{1}{x^q(x^{p-q}+1)}=\frac{1}{x^p(x^{q-p}+1)},\tag{1}$$ then there's two possible cases:
Case $p>q$: from the first equality of $(1)$ $$\frac{1}{x^p+x^q}\sim_0\frac{1}{x^q},$$ so $\displaystyle\int_0^1\frac{dx}{x^p+x^q}$ is convergent $\iff$ $q<1$, and from the second equality of $(1)$ $$\frac{1}{x^p+x^q}\sim_\infty\frac{1}{x^p},$$ so $\displaystyle\int_1^\infty\frac{dx}{x^p+x^q}$ is convergent $\iff$ $p>1$, hence $$\displaystyle\int_0^\infty\frac{dx}{x^p+x^q} \mathrm{is\, convergent} \iff 0\leq q<1<p.$$ Case $p=q$: it's easy to see that $\displaystyle\int_0^\infty\frac{dx}{x^p+x^q}$ is divergent.