Question Number 201464 by hardmath last updated on 06/Dec/23 $$\mathrm{if}\:\:\:\mathrm{f}\left(\mathrm{2}\right)\:=\:\mathrm{3}\:\:\:\mathrm{and}\:\:\:\mathrm{f}\left(\mathrm{4}\right)\:=\:\mathrm{5} \\ $$$$\mathrm{find}\:\:\:\int_{\mathrm{2}} ^{\:\mathrm{4}} \:\mathrm{f}\left(\mathrm{x}\right)\:\centerdot\:\mathrm{f}\:^{'} \left(\mathrm{x}\right)\:\mathrm{dx}\:=\:? \\ $$ Answered by mahdipoor last updated on 06/Dec/23 $$=\left[\frac{\left({f}\left({x}\right)\right)^{\mathrm{2}}…
Question Number 201462 by hardmath last updated on 06/Dec/23 $$\mathrm{Find}: \\ $$$$\int_{\mathrm{0}} ^{\:\boldsymbol{\pi}} \:\sqrt{\mathrm{1}\:-\:\mathrm{4}\:\mathrm{sin}^{\mathrm{2}} \:\frac{\mathrm{x}}{\mathrm{2}}\:\mathrm{cos}^{\mathrm{2}} \:\frac{\mathrm{x}}{\mathrm{2}}}\:\mathrm{dx}\:=\:? \\ $$ Answered by esmaeil last updated on 06/Dec/23…
Question Number 201460 by hardmath last updated on 06/Dec/23 $$\mathrm{Find}\:\mathrm{the}\:\mathrm{smallest}\:\mathrm{positive}\:\mathrm{period}\:\mathrm{of}\:\mathrm{the} \\ $$$$\mathrm{function}: \\ $$$$\mathrm{y}\:=\:\mid\:\mathrm{tan}\:\mathrm{2x}\:\mid\:\:+\:\:\mid\:\mathrm{cot}\:\mathrm{2x}\:\mid \\ $$ Answered by Mathspace last updated on 07/Dec/23 $${y}\left({x}\right)=\mid{tan}\left(\mathrm{2}{x}\right)\mid+\frac{\mathrm{1}}{\mid{tan}\left(\mathrm{2}{x}\right)\mid} \\…
Question Number 201430 by hardmath last updated on 06/Dec/23 $$\mathrm{Find}: \\ $$$$\frac{\mathrm{2}}{\mathrm{35}}\:+\:\frac{\mathrm{2}}{\mathrm{63}}\:+\:\frac{\mathrm{2}}{\mathrm{99}}\:+\:\frac{\mathrm{2}}{\mathrm{143}}\:=\:? \\ $$ Answered by Rasheed.Sindhi last updated on 06/Dec/23 $$\mathrm{2}\left(\frac{\mathrm{1}}{\mathrm{6}^{\mathrm{2}} −\mathrm{1}}+\frac{\mathrm{1}}{\mathrm{8}^{\mathrm{2}} −\mathrm{1}}+\frac{\mathrm{1}}{\mathrm{10}^{\mathrm{2}} −\mathrm{1}}+\frac{\mathrm{1}}{\mathrm{12}^{\mathrm{2}}…
Question Number 201463 by hardmath last updated on 06/Dec/23 $$\mathrm{a}\:=\:\mathrm{constant}\:\mathrm{number}: \\ $$$$\mathrm{if}\:\:\:\int\mathrm{x}\:\centerdot\:\mathrm{f}\left(\mathrm{x}\right)\:\mathrm{dx}\:=\:\mathrm{x}^{\mathrm{3}} \:-\:\mathrm{x}^{\mathrm{2}} \:+\:\mathrm{4x}\:-\:\frac{\mathrm{a}}{\mathrm{5}} \\ $$$$\mathrm{find}\:\:\:\mathrm{f}\left(\mathrm{2}\right)\:=\:? \\ $$ Answered by mr W last updated on…
Question Number 201441 by dimentri last updated on 06/Dec/23 $${Let}\:{f}\left({x}\right)\:{and}\:{g}\left({x}\right)\:{be}\:{given}\:{by}\: \\ $$$$\:{f}\left({x}\right)=\:\frac{\mathrm{1}}{{x}}\:+\frac{\mathrm{1}}{{x}−\mathrm{2}}\:+\frac{\mathrm{1}}{{x}−\mathrm{4}}\:+\:…\:+\frac{\mathrm{1}}{{x}−\mathrm{2018}} \\ $$$$\:{and}\: \\ $$$$\:\:{g}\left({x}\right)=\frac{\mathrm{1}}{{x}−\mathrm{1}}\:+\frac{\mathrm{1}}{{x}−\mathrm{3}}\:+\frac{\mathrm{1}}{{x}−\mathrm{5}}\:+…+\:\frac{\mathrm{1}}{{x}−\mathrm{2017}}. \\ $$$$\:\:{Prove}\:{that}\:\:\mid\:{f}\left({x}\right)−{g}\left({x}\right)\mid\:>\mathrm{2} \\ $$$$\:\:{for}\:{any}\:{non}−{integer}\:{real}\:{number} \\ $$$$\:\:{x}\:{satisfying}\:\mathrm{0}<{x}<\mathrm{2018}.\: \\ $$ Answered…
Question Number 201343 by gabi last updated on 05/Dec/23 Answered by aleks041103 last updated on 05/Dec/23 $${A}^{\mathrm{3}} +{A}^{\mathrm{2}} +{A}=\mathrm{0} \\ $$$$\lambda\:{is}\:{eigenvalue} \\ $$$$\Rightarrow\lambda^{\mathrm{3}} +\lambda^{\mathrm{2}} +\lambda=\lambda\left(\lambda^{\mathrm{2}}…
Question Number 201388 by MrGHK last updated on 05/Dec/23 Answered by witcher3 last updated on 05/Dec/23 $$=\underset{\mathrm{n}\geqslant\mathrm{1}} {\sum}\frac{\left(−\mathrm{1}\right)^{\mathrm{n}} }{\mathrm{n}}\underset{\mathrm{m}\geqslant\mathrm{0}} {\sum}\int_{\mathrm{0}} ^{\mathrm{1}} \left(−\mathrm{1}\right)^{\mathrm{m}} \mathrm{x}^{\mathrm{n}+\mathrm{2m}} \mathrm{dx} \\…
Question Number 201408 by Ari last updated on 05/Dec/23 Answered by mr W last updated on 05/Dec/23 $$=\frac{\mathrm{1}×\mathrm{3}}{\mathrm{2}^{\mathrm{2}} }×\frac{\mathrm{2}×\mathrm{4}}{\mathrm{3}^{\mathrm{2}} }×\frac{\mathrm{3}×\mathrm{5}}{\mathrm{4}^{\mathrm{2}} }×…×\frac{\mathrm{98}×\mathrm{100}}{\mathrm{99}^{\mathrm{2}} } \\ $$$$=\frac{\mathrm{1}×\cancel{\mathrm{3}}}{\mathrm{2}^{\cancel{\mathrm{2}}} }×\frac{\cancel{\mathrm{2}}×\cancel{\mathrm{4}}}{\cancel{\mathrm{3}^{\mathrm{2}}…
Question Number 201302 by hardmath last updated on 03/Dec/23 $$\begin{cases}{\mathrm{x}^{\mathrm{2}} \:+\:\mathrm{y}^{−\mathrm{2}} \:=\:\mathrm{4}}\\{\mathrm{x}^{−\mathrm{2}} \:+\:\mathrm{y}^{\mathrm{2}} \:=\:\mathrm{5}}\end{cases}\:\:\:\:\:\mathrm{find}:\:\:\frac{\mathrm{y}}{\mathrm{x}}\:=\:? \\ $$ Answered by witcher3 last updated on 03/Dec/23 $$\left(\mathrm{1}\right)\ast\left(\mathrm{2}\right)\Leftrightarrow\mathrm{2}+\left(\mathrm{xy}\right)^{\mathrm{2}} +\frac{\mathrm{1}}{\left(\mathrm{xy}\right)^{\mathrm{2}}…