Question Number 39015 by maxmathsup by imad last updated on 01/Jul/18
$${find}\:\:\int\:\:\:\:\:\:\frac{{dx}}{{x}\left(\mathrm{2}{x}+\mathrm{1}\right)\left(\mathrm{3}{x}+\mathrm{2}\right)} \\ $$$$\left.\mathrm{2}\right)\:{calculate}\:\:\int_{\mathrm{1}} ^{\mathrm{2}} \:\:\:\:\frac{{dx}}{{x}\left(\mathrm{2}{x}+\mathrm{1}\right)\left(\mathrm{3}{x}+\mathrm{2}\right)} \\ $$
Commented by math khazana by abdo last updated on 05/Jul/18
$${let}\:{decompose}\:{F}\left({x}\right)=\:\frac{\mathrm{1}}{{x}\left(\mathrm{2}{x}+\mathrm{1}\right)\left(\mathrm{3}{x}+\mathrm{2}\right)} \\ $$$${F}\left({x}\right)=\:\frac{{a}}{{x}}\:+\frac{{b}}{\mathrm{2}{x}+\mathrm{1}}\:+\frac{{c}}{\mathrm{3}{x}+\mathrm{2}} \\ $$$${a}\:={lim}_{{x}\rightarrow\mathrm{0}} {xF}\left({x}\right)=\frac{\mathrm{1}}{\mathrm{2}} \\ $$$${b}\:={lim}_{{x}\rightarrow−\frac{\mathrm{1}}{\mathrm{2}}} \:\:\left(\mathrm{2}{x}+\mathrm{1}\right){F}\left({x}\right)=\:\frac{\mathrm{1}}{\left(−\frac{\mathrm{1}}{\mathrm{2}}\right)\left(\frac{\mathrm{1}}{\mathrm{2}}\right)}\:=−\mathrm{4} \\ $$$${c}\:={lim}_{{x}\rightarrow−\frac{\mathrm{2}}{\mathrm{3}}} \:\:\left(−\mathrm{3}{x}+\mathrm{2}\right){F}\left({x}\right)=\:\frac{\mathrm{1}}{\left(−\frac{\mathrm{2}}{\mathrm{3}}\right)\left(−\frac{\mathrm{4}}{\mathrm{3}}+\mathrm{1}\right)} \\ $$$$=\:\frac{\mathrm{1}}{\left(−\frac{\mathrm{2}}{\mathrm{3}}\right)\left(−\frac{\mathrm{1}}{\mathrm{3}}\right)}\:=\frac{\mathrm{9}}{\mathrm{2}}\:\Rightarrow \\ $$$${F}\left({x}\right)=\:\frac{\mathrm{1}}{\mathrm{2}{x}}\:−\frac{\mathrm{4}}{\mathrm{2}{x}+\mathrm{1}}\:+\frac{\mathrm{9}}{\mathrm{2}\left(\mathrm{3}{x}+\mathrm{2}\right)}\:\Rightarrow \\ $$$$\int\:{F}\left({x}\right){dx}\:=\:\frac{\mathrm{1}}{\mathrm{2}}{ln}\mid{x}\mid\:−\mathrm{2}{ln}\mid\mathrm{2}{x}+\mathrm{1}\mid\:+\frac{\mathrm{3}}{\mathrm{2}}{ln}\mid\mathrm{3}{x}+\mathrm{2}\mid\:+{c} \\ $$
Commented by math khazana by abdo last updated on 05/Jul/18
![2) ∫_1 ^2 (dx/(x(2x+1)(3x+2))) =[(1/2)ln∣x∣ −2ln∣2x+1∣ +(3/2)ln∣3x+2∣]_1 ^2 =(1/2)ln(2)−2ln(5) +(3/2)ln(8) +2ln(3)−(3/2)ln(5)](https://www.tinkutara.com/question/Q39302.png)
$$\left.\mathrm{2}\right)\:\int_{\mathrm{1}} ^{\mathrm{2}} \:\:\:\frac{{dx}}{{x}\left(\mathrm{2}{x}+\mathrm{1}\right)\left(\mathrm{3}{x}+\mathrm{2}\right)} \\ $$$$=\left[\frac{\mathrm{1}}{\mathrm{2}}{ln}\mid{x}\mid\:−\mathrm{2}{ln}\mid\mathrm{2}{x}+\mathrm{1}\mid\:+\frac{\mathrm{3}}{\mathrm{2}}{ln}\mid\mathrm{3}{x}+\mathrm{2}\mid\right]_{\mathrm{1}} ^{\mathrm{2}} \\ $$$$=\frac{\mathrm{1}}{\mathrm{2}}{ln}\left(\mathrm{2}\right)−\mathrm{2}{ln}\left(\mathrm{5}\right)\:+\frac{\mathrm{3}}{\mathrm{2}}{ln}\left(\mathrm{8}\right)\:+\mathrm{2}{ln}\left(\mathrm{3}\right)−\frac{\mathrm{3}}{\mathrm{2}}{ln}\left(\mathrm{5}\right) \\ $$