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Question Number 72430 by mind is power last updated on 28/Oct/19

Hello find  finde  ∫_0 ^(+∞) ((ln(x))/(x^2 +ax+b))dx  conditions a^2 <4b     in therm of x_1 ,x_2   root of X^2 +aX+b    hint Residus theorem applied too ((log^2 (z))/(z^2 +az+b))  this is very usufull i find it in lecture yesterday  because we can easly evaluat any kinde of ∫_0 ^(+∞) ((log^k (z))/(p(z)))dz  withe p(z) eiwthout  root in ]0,+∞[ deg(p(z))≥2

$$\mathrm{Hello}\:\mathrm{find} \\ $$ $$\mathrm{finde}\:\:\int_{\mathrm{0}} ^{+\infty} \frac{\mathrm{ln}\left(\mathrm{x}\right)}{\mathrm{x}^{\mathrm{2}} +\mathrm{ax}+\mathrm{b}}\mathrm{dx} \\ $$ $$\mathrm{conditions}\:\mathrm{a}^{\mathrm{2}} <\mathrm{4b}\:\:\: \\ $$ $$\mathrm{in}\:\mathrm{therm}\:\mathrm{of}\:\mathrm{x}_{\mathrm{1}} ,\mathrm{x}_{\mathrm{2}} \:\:\mathrm{root}\:\mathrm{of}\:\mathrm{X}^{\mathrm{2}} +\mathrm{aX}+\mathrm{b}\: \\ $$ $$\:\mathrm{hint}\:\mathrm{Residus}\:\mathrm{theorem}\:\mathrm{applied}\:\mathrm{too}\:\frac{\mathrm{log}^{\mathrm{2}} \left(\mathrm{z}\right)}{\mathrm{z}^{\mathrm{2}} +\mathrm{az}+\mathrm{b}} \\ $$ $$\mathrm{this}\:\mathrm{is}\:\mathrm{very}\:\mathrm{usufull}\:\mathrm{i}\:\mathrm{find}\:\mathrm{it}\:\mathrm{in}\:\mathrm{lecture}\:\mathrm{yesterday} \\ $$ $$\mathrm{because}\:\mathrm{we}\:\mathrm{can}\:\mathrm{easly}\:\mathrm{evaluat}\:\mathrm{any}\:\mathrm{kinde}\:\mathrm{of}\:\int_{\mathrm{0}} ^{+\infty} \frac{\mathrm{log}^{\mathrm{k}} \left(\mathrm{z}\right)}{\mathrm{p}\left(\mathrm{z}\right)}\mathrm{dz} \\ $$ $$\left.\mathrm{withe}\:\mathrm{p}\left(\mathrm{z}\right)\:\mathrm{eiwthout}\:\:\mathrm{root}\:\mathrm{in}\:\right]\mathrm{0},+\infty\left[\:\mathrm{deg}\left(\mathrm{p}\left(\mathrm{z}\right)\right)\geqslant\mathrm{2}\right. \\ $$

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