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$\int_{0}^{+\infty}\frac{\log^2{x}}{(x+\alpha)(x+\beta)}\,dx$

Calculus (Integrals, Series)
Grigorios Kostakos
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$\int_{0}^{+\infty}\frac{\log^2{x}}{(x+\alpha)(x+\beta)}\,dx$

As a continuation of this:

For $0<\alpha<\beta$, evaluate
$\displaystyle\int_{0}^{+\infty}\frac{\log^2{x}}{(x+\alpha)(x+\beta)}\,dx\,.$
Grigorios Kostakos
Grigorios Kostakos
Founder
Articles: 0
Posts: 460
Joined: Mon Nov 09, 2015 1:36 am
Location: Ioannina, Greece

Re: $\int_{0}^{+\infty}\frac{\log^2{x}}{(x+\alpha)(x+\beta)}\,dx$

For the sake of discussion I give the next step:

Considering the complex function $$f(z)=\dfrac{{\rm{Log}}^3{z}+\pi^2{\rm{Log}}{z}}{(z-\alpha)(z-\beta)}\,,\quad z\in\mathbb{C}\setminus\{x+0i\;|\;x\leqslant0\},$$ by the same procedure as in $\int_{0}^{+\infty}\frac{\log{x}}{(x+\alpha)(x+\beta)}\,dx$, we will get
$\displaystyle\int_{0}^{+\infty}\frac{\log^2{x}}{(x+\alpha)(x+\beta)}\,dx=\dfrac{\log^3\beta+\pi^2\log\beta-\log^3\alpha-\pi^2\log\alpha}{3\,(\beta-\alpha)}\,.$
Can you find appropriate complex functions for higher powers of $\log{x}$ i.e. $\log^3{x}\,,\; \log^4{x}\,,$ ...?
Grigorios Kostakos