How would I go about finding values of $a$ and $b$ such that something like:
$\int_{5}^{\infty} \frac{x^a}{\ln^bx} dx$
is convergent?
Is there a standard/systematic approach to going about doing this or anything of the like? At the moment, I'm just testing values in online resources to see which combinations work but I'm guessing and pretty certain that there is a better way to go about this.
You can handle this easily if you first make the substitution $y=\ln x$. The integral becomes $\int_{\ln 5}^{\infty} \frac {e^{(a+1)y}} {y^{b}}$. If $a<-1$ the integral clearly converges. [ Because $\frac 1 {y^{b}}$ is bounded]. If $a>-1$ it diverges. Here you use the principle that if $c>0$ then $e^{cy} \to \infty$ 'faster' than any power of $y$ so even though the numerator and the denominator both $\to \infty$ it is the numerator which 'counts'. [ A rigorous justification can be given by showing that $\frac {e^{cy}} {y^{b}} \geq e^{cy/2}$ for $y$ sufficiently large]. Now you are left with the case $a=-1$. I leave it to you to see when $\int_{\ln 5}^{\infty} \frac 1 {y^{b}}<\infty$.