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.
$\endgroup$ 13 Answers
$\begingroup$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.
$\endgroup$ 2 $\begingroup$Hint: $x^a$ is integrable near $x=0$ when $a>-1$ and $x^b$ is integrable as $x \rightarrow \infty$ when $b < -1$.
$$\int_0^r dx \: x^a = \frac{r^{a+1} - 0^{a+1}}{a+1}$$
Note that if $a+1<0$, then the numerator is infinite (i.e., 1/0). If $a+1=0$, then the denominator is zero. So $a+1>0$ for convergence at $x=0$.
$$\lim_{s \rightarrow \infty}\int_r^{s} dx \: x^b= \lim_{s \rightarrow \infty}\frac{s^{b+1}-r^{b+1}}{b+1}$$
For the limit to exist in the numerator, $b+1<0$. If $b+1=0$, then the denominator is infinite. So $b+1<0$ for convergence at infinity.
$\endgroup$ 2 $\begingroup$Noting some valuable points as hints besides to @Ron's answer:
$\int_{0}^{+\infty} \frac{dx}{x^p + x^q}=\int_{0}^{1} \frac{dx}{x^p + x^q}+\int_{1}^{+\infty} \frac{dx}{x^p + x^q}$
$\int_a^bf(x)dx$ converges if $p<1$ and $\lim_{x\to a^+}(x-a)^pf(x)=A$ is finite.
$\int_a^bf(x)dx$ diverges if $p\geq1$ and $A\neq 0$ ($A$ may be infinite).
$\int_a^{\infty}f(x)dx$ converges if $p>1$ and $\lim_{x\to a^+}x^pf(x)=A$ is finite.
$\int_a^{\infty}f(x)dx$ diverges if $p\leq1$ and $A\neq 0$ ($A$ may be infinite).
