Question Number 221135 by MathematicalUser2357 last updated on 25/May/25 $$\mathrm{South}\:\mathrm{Korean}\:\mathrm{Grade}\:\mathrm{12}\:\mathrm{math} \\ $$$$\mathrm{Prove}\:\mathrm{log}_{{a}} {M}^{{n}} ={n}\mathrm{log}_{{a}} {M} \\ $$$$\mathrm{Using}\:\mathrm{below}: \\ $$$$\mathrm{When}\:{M}={a}^{{x}} ,\:\mathrm{log}_{{a}} {M}={x} \\ $$$$\mathrm{When}\:{N}={a}^{{y}} ,\:\mathrm{log}_{{a}} {N}={y}…
Question Number 221129 by SdC355 last updated on 25/May/25 $$\mathrm{prove} \\ $$$$\mathrm{Contour}\:\mathrm{integral}\:\mathrm{repreasentation} \\ $$$$\begin{pmatrix}{{p}}\\{{q}}\end{pmatrix}=\frac{\mathrm{1}}{\mathrm{2}\pi\boldsymbol{{i}}}\:\oint_{\:{C}} \:\left(\mathrm{1}−{z}\right)^{{p}} {z}^{−{q}} \:\frac{\mathrm{d}{z}}{{z}} \\ $$ Answered by MrGaster last updated on…
Question Number 221153 by gregori last updated on 25/May/25 $$\:\:{f}\left({x}\right)=\:\frac{\mathrm{1}}{\mathrm{2}^{{x}} }\:+\:\frac{\mathrm{1}}{\mathrm{3}^{{x}} }\:+\:\frac{\mathrm{1}}{\mathrm{4}^{{x}} }\:+\:…\:+\frac{\mathrm{1}}{\mathrm{4000}^{{x}} } \\ $$$$\:\:{f}\left(\mathrm{2}\right)\:+\:{f}\left(\mathrm{3}\right)\:+\:{f}\left(\mathrm{4}\right)+\:…\:=? \\ $$ Commented by Frix last updated on 25/May/25…
Question Number 221154 by gregori last updated on 25/May/25 $$\:{f}\left({x}\right)=\:\frac{{x}}{\mid\:{x}\:\mid\:+\:\mathrm{1}} \\ $$$$\:\:{f}\left({f}\left({f}\left({f}\left({x}\right)\right)\right)\right)\:=? \\ $$ Commented by Frix last updated on 25/May/25 $$\frac{{x}}{\mathrm{4}\mid{x}\mid+\mathrm{1}} \\ $$$${f}_{\mathrm{1}} \left({x}\right)=\frac{{x}}{\mid{x}\mid+\mathrm{1}}…
Question Number 221081 by SdC355 last updated on 24/May/25 $$\mathrm{for}\:\mathrm{all}\:{m},{n},{p}\in\mathbb{R} \\ $$$$\left(\mathrm{g}\left({m}\right)+{n}\right)\left(\mathrm{g}\left({n}\right)+{m}\right)={p}^{\mathrm{2}} \left(\mathrm{perfect}\:\mathrm{square}\:\mathrm{number}\right) \\ $$$$\mathrm{find}\:\mathrm{function}\:\mathrm{g};\mathbb{N}\rightarrow\mathbb{N} \\ $$ Terms of Service Privacy Policy Contact: info@tinkutara.com
Question Number 221078 by MrGaster last updated on 24/May/25
Question Number 221100 by Nicholas666 last updated on 24/May/25 $$ \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\boldsymbol{\mathrm{Prove}}; \\ $$$$\:\:\int_{\mathrm{0}} ^{\:+\infty} \:\frac{\mathrm{4}\centerdot\boldsymbol{\mathrm{cos}}\:\boldsymbol{{x}}\:\centerdot\:\sqrt[{\mathrm{6}\:\:}]{\boldsymbol{\mathrm{sinh}}\:\boldsymbol{{x}}\:}}{\boldsymbol{\mathrm{sinh}}\:\boldsymbol{{x}}\:+\:\boldsymbol{\mathrm{sinh}}\:\mathrm{3}\boldsymbol{{x}}\:+\:\mathrm{4}\:\boldsymbol{\mathrm{sinh}}^{\mathrm{2}} \:\boldsymbol{{x}}\:−\:\mathrm{2}\:\boldsymbol{\mathrm{sinh}}^{\mathrm{2}} \:\mathrm{2}\boldsymbol{{x}}\:+\:\mathrm{4}\:\boldsymbol{\mathrm{sinh}}^{\mathrm{4}} \:\boldsymbol{{x}}\:+\:\mathrm{4}\:\boldsymbol{\mathrm{cosh}}^{\mathrm{4}} \boldsymbol{{x}}}\:\boldsymbol{\mathrm{d}{x}}\:=\:\frac{\boldsymbol{\pi}}{\:\sqrt{\mathrm{6}}\:+\:\mathrm{2}}\:\:\:\:\:\:\: \\ $$$$ \\ $$ Terms…
Question Number 221102 by fantastic last updated on 24/May/25 Answered by mehdee7396 last updated on 25/May/25 $${AB}=\mathrm{2}\sqrt{{ar}}\:\:\&\:\:{BC}=\mathrm{2}\sqrt{{br}}\:\:\:\&\:\:\:{AC}=\mathrm{2}\sqrt{{ab}} \\ $$$$\Rightarrow\sqrt{{ab}}=\left(\sqrt{{a}}+\sqrt{{b}}\right)\sqrt{{r}\:} \\ $$$$\Rightarrow{r}=\frac{{ab}}{{a}+{b}+\mathrm{2}\sqrt{{ab}}}\:\:{or}\:\:\frac{\mathrm{1}}{\:\sqrt{{r}}}=\frac{\mathrm{1}}{\:\sqrt{{a}}}+\frac{\mathrm{1}}{\:\sqrt{{b}}}\: \\ $$$$ \\ $$…
Question Number 221103 by Jgrads last updated on 24/May/25 $$\mathrm{Prove}\::\:\:\:\:\:\forall\mathrm{x}\in\mathrm{IR},\:\forall\mathrm{n}\in\mathrm{IN}^{\ast} \: \\ $$$$\underset{\:\mathrm{0}} {\int}^{\:\frac{\pi}{\mathrm{2}}} \mathrm{ch}\left(\mathrm{2xt}\right)\mathrm{cos}^{\mathrm{2n}} \left(\mathrm{t}\right)\:\mathrm{dt}\:\leqslant\:\mathrm{e}^{\frac{\mathrm{x}^{\mathrm{2}} }{\mathrm{n}}} \underset{\:\mathrm{0}} {\int}^{\:\frac{\pi}{\mathrm{2}}} \mathrm{cos}^{\mathrm{2n}} \left(\mathrm{t}\right)\:\mathrm{dt} \\ $$ Terms of…
Question Number 221099 by Nicholas666 last updated on 24/May/25 $$ \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\mathrm{Prove}: \\ $$$$\:\:\underset{\:−\pi} {\overset{\:\pi} {\int}}\:\underset{{n}=\mathrm{0}} {\overset{\infty} {\sum}}\:\frac{\mathrm{cos}^{{n}\:+\:\mathrm{1}} \:{x}}{\left({n}\:+\:\mathrm{1}\right)\left(\mathrm{1}\:+\:{e}^{{x}^{\mathrm{2}{n}\:+\mathrm{1}} } \right)}\:\:\mathrm{d}{x}\:=\:\pi\:\mathrm{ln2}\:\:\:\:\: \\ $$$$ \\ $$…