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x-4y-2x-2-9xy-dx-M-m-




Question Number 183101 by Mastermind last updated on 20/Dec/22
∫((x+4y)/(2x^2 +9xy))dx      M.m
$$\int\frac{\mathrm{x}+\mathrm{4y}}{\mathrm{2x}^{\mathrm{2}} +\mathrm{9xy}}\mathrm{dx} \\ $$$$ \\ $$$$ \\ $$$$\mathrm{M}.\mathrm{m} \\ $$
Commented by mr W last updated on 20/Dec/22
is y a constant or a function of x?
$${is}\:{y}\:{a}\:{constant}\:{or}\:{a}\:{function}\:{of}\:{x}? \\ $$
Commented by MJS_new last updated on 21/Dec/22
wrong.  maybe better celebrate after learning the  basics?!
$$\mathrm{wrong}. \\ $$$$\mathrm{maybe}\:\mathrm{better}\:\mathrm{celebrate}\:{after}\:\mathrm{learning}\:\mathrm{the} \\ $$$$\mathrm{basics}?! \\ $$
Commented by mr W last updated on 21/Dec/22
even maybe better learn basic human  behavior at first before leraning basic  mathematics!  i kindly asked him many times not   to post answers as “comment” but as  “answer”, i even told him how to do  this, i even told him in his language  (French), but he ignored all those   kindnesses and even didn′t give any  reply. therefore i would say what he   mostly misses is the basic human   behavior.
$${even}\:{maybe}\:{better}\:{learn}\:{basic}\:{human} \\ $$$${behavior}\:{at}\:{first}\:{before}\:{leraning}\:{basic} \\ $$$${mathematics}! \\ $$$${i}\:{kindly}\:{asked}\:{him}\:{many}\:{times}\:{not}\: \\ $$$${to}\:{post}\:{answers}\:{as}\:“{comment}''\:{but}\:{as} \\ $$$$“{answer}'',\:{i}\:{even}\:{told}\:{him}\:{how}\:{to}\:{do} \\ $$$${this},\:{i}\:{even}\:{told}\:{him}\:{in}\:{his}\:{language} \\ $$$$\left({French}\right),\:{but}\:{he}\:{ignored}\:{all}\:{those}\: \\ $$$${kindnesses}\:{and}\:{even}\:{didn}'{t}\:{give}\:{any} \\ $$$${reply}.\:{therefore}\:{i}\:{would}\:{say}\:{what}\:{he}\: \\ $$$${mostly}\:{misses}\:{is}\:{the}\:{basic}\:{human}\: \\ $$$${behavior}. \\ $$
Answered by MJS_new last updated on 21/Dec/22
∫((x+4y)/(2x^2 +9xy))dx=∫((1/(9(2x+9y)))+(4/(9x)))dx=  =(1/(18))ln ∣2x+9y∣ +(4/9)ln ∣x∣ +C
$$\int\frac{{x}+\mathrm{4}{y}}{\mathrm{2}{x}^{\mathrm{2}} +\mathrm{9}{xy}}{dx}=\int\left(\frac{\mathrm{1}}{\mathrm{9}\left(\mathrm{2}{x}+\mathrm{9}{y}\right)}+\frac{\mathrm{4}}{\mathrm{9}{x}}\right){dx}= \\ $$$$=\frac{\mathrm{1}}{\mathrm{18}}\mathrm{ln}\:\mid\mathrm{2}{x}+\mathrm{9}{y}\mid\:+\frac{\mathrm{4}}{\mathrm{9}}\mathrm{ln}\:\mid{x}\mid\:+{C} \\ $$

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