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Question-227273

Question Number 227273 by Spillover last updated on 11/Jan/26 Answered by peace2 last updated on 12/Jan/26 $$\frac{\mathrm{1}}{{y}_{} }={z} \\ $$$$\Rightarrow\frac{{dz}}{{dx}}=−\frac{{dy}}{{dx}}.\frac{\mathrm{1}}{{y}^{\mathrm{2}} } \\ $$$$\Leftrightarrow−{x}\frac{{dz}}{{dx}}+{z}={ln}\left({x}\right) \\ $$$${z}\left({x}\right)={kx}\Rightarrow−{x}\left({k}'{x}\right)={ln}\left({x}\right)\Rightarrow{k}'=−\frac{{ln}\left({x}\right)}{{x}^{\mathrm{2}}…

Does-condition-sup-x-R-f-1-x-lt-Gaurantee-f-x-is-Bounded-in-R-

Question Number 227274 by Lara2440 last updated on 11/Jan/26 $$\mathrm{Does}\:\mathrm{condition} \\ $$$$\underset{{x}\in\mathbb{R}} {\mathrm{sup}}\:\mid\mid{f}^{\left(\mathrm{1}\right)} \left({x}\right)\mid\mid<\infty\:,\:\:\mathrm{Gaurantee}\:{f}\left({x}\right)\:\mathrm{is}\:\mathrm{Bounded}\:\mathrm{in}\:\mathbb{R}? \\ $$ Answered by peace2 last updated on 12/Jan/26 $${non}\:{f}\left({x}\right)={x} \\…

Question-227271

Question Number 227271 by Spillover last updated on 11/Jan/26 Answered by Kassista last updated on 11/Jan/26 $$ \\ $$$$\int\:\frac{{x}^{\mathrm{2}} }{{x}^{\mathrm{2}} +\mathrm{5}}\:{dx}\:=\:\int\:\frac{{x}^{\mathrm{2}} +\mathrm{5}−\mathrm{5}}{{x}^{\mathrm{2}} +\mathrm{5}}\:{dx}\:=\:\int\:\frac{{x}^{\mathrm{2}} +\mathrm{5}}{{x}^{\mathrm{2}} +\mathrm{5}}\:{dx}\:−\int\frac{\mathrm{5}}{{x}^{\mathrm{2}}…

Proof-that-2-is-irrational-

Question Number 227283 by Edwanrosario201444 last updated on 11/Jan/26 $${Proof}\:{that}\:\sqrt{\mathrm{2}}\:{is}\:{irrational} \\ $$ Answered by TonyCWX last updated on 11/Jan/26 $$\mathrm{Suppose}\:\sqrt{\mathrm{2}}\:\mathrm{is}\:\mathrm{rational} \\ $$$$\mathrm{Let}\:\mathrm{a}\:\mathrm{and}\:\mathrm{b}\:\mathrm{be}\:\mathrm{positive}\:\mathrm{integers}\:\mathrm{such}\:\mathrm{that} \\ $$$$\sqrt{\mathrm{2}}\:=\:\frac{\mathrm{a}}{\mathrm{b}} \\…