Question Number 226771 by Spillover last updated on 14/Dec/25 Answered by Frix last updated on 14/Dec/25 $${a}\neq\pm\mathrm{1} \\ $$$$\mathrm{We}\:\mathrm{can}\:\mathrm{say}\:{a}>\mathrm{0}\:\mathrm{because} \\ $$$$\underset{\mathrm{0}} {\overset{\pi} {\int}}\frac{{d}\theta}{\mathrm{1}−\mathrm{2}{a}\mathrm{cos}\:\theta\:+{a}^{\mathrm{2}} }=\underset{\mathrm{0}} {\overset{\pi}…
Question Number 226702 by leromain last updated on 10/Dec/25 $$\int\frac{{ln}\left({x}^{\mathrm{2}} +\mathrm{3}{x}+\mathrm{2}\right)}{{x}^{\mathrm{2}} +\mathrm{1}}{dx}\left[\right. \\ $$ Answered by Frix last updated on 11/Dec/25 $$\int\frac{\mathrm{ln}\:\left({x}^{\mathrm{2}} +\mathrm{3}{x}+\mathrm{2}\right)}{{x}^{\mathrm{2}} +\mathrm{1}}{dx}=\int\frac{\mathrm{ln}\:\left(\left({x}+\mathrm{1}\right)\left({x}+\mathrm{2}\right)\right)}{{x}^{\mathrm{2}} +\mathrm{1}}{dx}=…
Question Number 224615 by necx122 last updated on 21/Sep/25 $${The}\:{vectors}\:{OP},\:{OQ}\:{and}\:{OR}\:{represented} \\ $$$${by}\:{a},{b}\:{and}\:{c}\:{respectively}:\:{where}\:{a}=\mathrm{10}{i}+{j}, \\ $$$${b}=−\mathrm{2}{i}+\mathrm{7}{j},\:{c}={a}+\mathrm{3}{b},\:{and}\:{O}\:{is}\:{the}\:{origin}. \\ $$$${OR}\:{and}\:{PR}\:{intersects}\:{at}\:{M}\:{where} \\ $$$${OM}={kOR}\:{and}\:{PM}={lPQ}\:{and}\:{k},\:{l}\:{are} \\ $$$${constants}.\:{Find}: \\ $$$$\left({i}\right)\:{The}\:{equation}\:{of}\:{the}\:{lines}\:{of}\:{PQ}\:{and} \\ $$$${OR} \\…
Question Number 224224 by math0101 last updated on 25/Aug/25 Commented by Rasheed.Sindhi last updated on 26/Aug/25 $${What}\:{are}\:{the}\:{definitions}\:{of}\:\square\:{and}\:\centerdot\:\:? \\ $$ Commented by fkwow344 last updated on…
Question Number 222453 by Nicholas666 last updated on 27/Jun/25 $$ \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\int_{\mathrm{0}} ^{\:\infty} \:\frac{\mathrm{tanh}^{\mathrm{2}} \left({x}\right)}{{x}^{\mathrm{2}} }\:{dx} \\ $$$$ \\ $$ Answered by MrGaster last updated…
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Question Number 219658 by Nicholas666 last updated on 30/Apr/25 $$ \\ $$$$\:\:\:\:\:\:\:\:\:\:\:{prove} \\ $$$$\:\:\:\underset{{n}=\mathrm{2}} {\overset{\infty} {\prod}}\:{e}\left(\mathrm{1}−\frac{\mathrm{1}}{{n}^{\mathrm{2}} }\overset{{n}^{\mathrm{2}} } {\right)}=\:\frac{\pi}{{e}\sqrt{{e}}} \\ $$$$ \\ $$ Answered by…
Question Number 219341 by alcohol last updated on 23/Apr/25 $$\int\frac{{dx}}{\mathrm{1}\:+\:{sin}^{\mathrm{3}} {x}\:+\:{cos}^{\mathrm{3}} {x}} \\ $$ Commented by Ghisom last updated on 23/Apr/25 $$\mathrm{simply}\:\mathrm{use}\:{t}=\mathrm{tan}\:\frac{{x}}{\mathrm{2}}\:\mathrm{and}\:\mathrm{decompose} \\ $$$$\mathrm{which}\:\mathrm{leads}\:\mathrm{to} \\…
Question Number 219262 by amresh last updated on 21/Apr/25 $$\overset{} {\mathrm{E}lectric}\:\mathrm{field}\:\mathrm{strenth}\:\mathrm{at}\:\mathrm{any}\:\mathrm{point}\:\mathrm{in}\:\mathrm{the}\:\mathrm{space} \\ $$$$\mathrm{is}\:\mathrm{defined}\:\mathrm{as}\:\mathrm{the}\:\mathrm{force}\:\mathrm{per}\:\mathrm{unit}\:\mathrm{charge}\:\mathrm{at}\:\mathrm{that}\:\mathrm{point}. \\ $$$$\:\mathrm{It}\:\mathrm{is}\:\mathrm{a}\:\mathrm{vector}\:\mathrm{quantity}\:\mathrm{whose}\:\mathrm{magnitude}\:\mathrm{is} \\ $$$$\mathrm{given}\:\mathrm{by}\:\mathrm{Coulomb}^{\mathrm{s}\:\:} \:\mathrm{law}\:\mathrm{and}\:\mathrm{diection}\:\mathrm{is}\:\mathrm{in}\: \\ $$$$\mathrm{straight}\:\mathrm{line}\:\mathrm{loining}\:\mathrm{the}\:\mathrm{at}\:\mathrm{that}\:\mathrm{point}. \\ $$$$\mathrm{mathemstically} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\: \\…
Question Number 219185 by fantastic last updated on 20/Apr/25 Answered by MrGaster last updated on 20/Apr/25 $$\bigtriangledown\centerdot\left(\overset{\rightarrow} {{F}}×\overset{\rightarrow} {{G}}\right)=\partial_{{i}} \left(\epsilon_{{ijk}} {F}_{{j}} {G}_{{k}} \right) \\ $$$$=\epsilon_{{ijk}}…