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Author: Tinku Tara

ln-x-4-1-ln-x-2-1-3-2-dx-

Question Number 152866 by talminator2856791 last updated on 02/Sep/21 $$\: \\ $$$$\: \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\int_{−\infty} ^{\:\infty} \:\frac{\mathrm{ln}\left(\sqrt{{x}^{\mathrm{4}} +\mathrm{1}}\right)}{\left(\mathrm{ln}\left(\left(\sqrt{{x}^{\mathrm{2}} +\mathrm{1}}\right)^{\mathrm{3}} \right)\right)^{\mathrm{2}} }\:\:{dx} \\ $$$$\: \\ $$$$\: \\…

ln-x-x-2-1-1-2-1-ln-x-2-1-x-x-1-dx-

Question Number 152861 by talminator2856791 last updated on 02/Sep/21 $$\: \\ $$$$\:\:\:\:\:\:\: \\ $$$$\int_{−\infty} ^{\:\infty} \frac{\left(\mathrm{ln}\left(\left({x}^{\sqrt{{x}^{\mathrm{2}} +\mathrm{1}}} +\mathrm{1}\right)^{\mathrm{2}} +\mathrm{1}\right)\right)^{−\mathrm{ln}\left({x}^{\mathrm{2}} +\mathrm{1}\right)} }{\:\sqrt{{x}^{\mid\lfloor{x}\rfloor\mid} +\mathrm{1}}}\:\:{dx} \\ $$$$\: \\…

1-e-x-dx-2-sin-x-sin-x-cos-x-dx-

Question Number 87325 by M±th+et£s last updated on 04/Apr/20 $$\left.\mathrm{1}\right)\:\int{e}^{\sqrt{{x}}} \:{dx} \\ $$$$\left.\mathrm{2}\right)\int\frac{\sqrt{{sin}\left({x}\right)}}{\:\sqrt{{sin}\left({x}\right)}+\sqrt{{cos}\left({x}\right)}}{dx} \\ $$ Commented by Serlea last updated on 04/Apr/20 $$\left.\mathrm{1}\right)\:\int\mathrm{e}^{\sqrt{\mathrm{x}}} \mathrm{dx} \\…

Monsieur-Puissant-je-quitte-ce-forum-mathe-matique-de-finitivement-mais-sans-avoir-dit-que-j-ai-adore-e-changer-avec-vous-Bonne-continuation-et-vive-les-maths-

Question Number 152856 by Olaf_Thorendsen last updated on 02/Sep/21 $$\mathrm{Monsieur}\:\mathrm{Puissant},\:\mathrm{je}\:\mathrm{quitte}\:\mathrm{ce}\:\mathrm{forum} \\ $$$$\mathrm{math}\acute {\mathrm{e}matique}\:\mathrm{d}\acute {\mathrm{e}finitivement}\:\mathrm{mais} \\ $$$$\mathrm{sans}\:\mathrm{avoir}\:\mathrm{dit}\:\mathrm{que}\:\mathrm{j}'\mathrm{ai}\:\mathrm{ador}\acute {\mathrm{e}}\:\acute {\mathrm{e}changer} \\ $$$$\mathrm{avec}\:\mathrm{vous}. \\ $$$$ \\ $$$$\mathrm{Bonne}\:\mathrm{continuation}\:\mathrm{et}\:\mathrm{vive}\:\mathrm{les} \\…

Call-a-positive-integer-n-good-if-there-are-n-integers-positive-or-negative-and-not-necessarily-distinct-such-that-their-sum-and-product-are-both-equal-to-n-e-g-8-is-good-since-8-4-2-1-1-1-1-1-

Question Number 21784 by Tinkutara last updated on 03/Oct/17 $$\mathrm{Call}\:\mathrm{a}\:\mathrm{positive}\:\mathrm{integer}\:{n}\:\boldsymbol{\mathrm{good}}\:\mathrm{if}\:\mathrm{there} \\ $$$$\mathrm{are}\:{n}\:\mathrm{integers},\:\mathrm{positive}\:\mathrm{or}\:\mathrm{negative},\:\mathrm{and} \\ $$$$\mathrm{not}\:\mathrm{necessarily}\:\mathrm{distinct},\:\mathrm{such}\:\mathrm{that}\:\mathrm{their} \\ $$$$\mathrm{sum}\:\mathrm{and}\:\mathrm{product}\:\mathrm{are}\:\mathrm{both}\:\mathrm{equal}\:\mathrm{to}\:{n} \\ $$$$\left(\mathrm{e}.\mathrm{g}.\:\mathrm{8}\:\mathrm{is}\:\boldsymbol{\mathrm{good}}\:\mathrm{since}\right. \\ $$$$\mathrm{8}=\mathrm{4}\centerdot\mathrm{2}\centerdot\mathrm{1}\centerdot\mathrm{1}\centerdot\mathrm{1}\centerdot\mathrm{1}\left(−\mathrm{1}\right)\left(−\mathrm{1}\right)=\mathrm{4}+\mathrm{2}+\mathrm{1}+\mathrm{1}+\mathrm{1} \\ $$$$\left.+\mathrm{1}+\left(−\mathrm{1}\right)+\left(−\mathrm{1}\right)\right). \\ $$$$\mathrm{Show}\:\mathrm{that}\:\mathrm{integers}\:\mathrm{of}\:\mathrm{the}\:\mathrm{form}\:\mathrm{4}{k}\:+\:\mathrm{1} \\…