Let $a, b, c, d \in \mathbb{R}$. I was wondering does the integreal $\int_A^{\infty} \frac{1}{(ax+b)(cx+d)} dx$ converge? where the integrand is well defined for $x\geq A$?

I think it should because $1/x^2$ does... but I wasn't sure. Any info would be appreciated!

  • $\begingroup$ As long as the domain over which you integrate does not include the roots to $(ax+b)(cx+d)=0$ $\endgroup$ – Henry Lee May 12 at 19:03

Yes, it converges, since$$\left\lvert\frac1{(ax+b)(cx+d)}\right\rvert\leqslant\frac2{\lvert ac\rvert x^2}$$if $x$ is large enough and because $\int_A^\infty\frac{\mathrm dx}{x^2}$ converges.

  • $\begingroup$ Nitpick: your inequality only holds for positive parameters. $\endgroup$ – Alexander Geldhof May 12 at 8:39
  • $\begingroup$ No, it is not a nitpick. I've edited my answer. Thank you. $\endgroup$ – José Carlos Santos May 12 at 8:41

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