For which $p$, $q$ does the improper integral exist?

Question

For which $p$, $q$ does the improper integral exist? $$ \text{(a)}\quad \int_0^\infty \frac{dx}{x^p+x^q} \qquad\quad\text{(b)}\quad \int_0^\infty x^p |x-1|^q \, dx $$

I tried for (a) to simplify the problem

• if $p=q$ it doesn't exist

I am not sure if that's true. I am also having hard time solving (b). I'd be thankful if you helped me with (b) at least.

Answer 1

You should use the asymptotic comparison test for improper integrals (no explicit integration is needed).

As regards (a), we have that, as $x\to 0^+$, $$ \frac{1}{x^p+x^q}\sim\frac{1}{x^{\min(p,q)}}$$ and the integral is finite in $(0,a)$ iff $\min(p,q)<1$.

On the other hand as $x\to +\infty$, $$ \frac{1}{x^p+x^q}\sim\frac{1}{x^{\max(p,q)}}$$ and the integral is finite in $(a,+\infty)$ iff $\max(p,q)>1$.

So we may conclude that the whole integral over $(0,+\infty)$ is convergent if and only if $$\min(p,q)<1\quad\text{AND}\quad \max(p,q)>1.$$

Now try (b) and note that in this case you should investigate what happens near $0^+$, $+\infty$ AND $1$.

i) if $x\to 0^+$ then $$x^p |x-1|^q \sim x^p\implies ?$$

ii) if $x\to 1$ then $$x^p |x-1|^q \sim |x-1|^q\implies ?$$

iii) if $x\to +\infty$ then $$x^p |x-1|^q \sim x^{p+q}\implies ?$$

Answer 2

Here is case (a). For it to converge, we have to have $\text{max}\{p,q\}>1$ and $\text{min}\{p,q\}<1$ Let $p-q=\alpha$.

\begin{align} \int_0^\infty \frac{1}{x^p+x^q}\,dx&=\int_0^\infty \frac{x^{-q}}{1+x^{p-q}}\,dx \\ &=\frac{1}{\alpha}\int_0^\infty \frac{u^{-q/\alpha}}{1+u} u^{1/\alpha-1}\,du \\ &=\frac{1}{\alpha}\int_0^\infty \frac{u^{(1-p)/\alpha}}{1+u}\,du \\ &=\frac{1}{\alpha}\frac{\Gamma\big(\frac{1-q}{\alpha}\big)\Gamma\big(\frac{p-1}{\alpha}\big)}{\Gamma\big(1\big)} \tag{1}\\ &=\frac{1}{\alpha}\Gamma\left(\frac{1-q}{\alpha}\right)\,\Gamma\left(\frac{p-1}{\alpha}\right) \\ \\ &=\frac{1}{p-q}\Gamma\left(\frac{1-q}{p-q}\right)\,\Gamma\left(\frac{p-1}{p-q}\right) \end{align} Where I've used an integral representation of the Beta function, and its relationship with the Gamma function in $(1)$.