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Question Number 219869 Answers: 1 Comments: 0
Question Number 219868 Answers: 1 Comments: 0
$$ \\ $$$$\:\:\:\:\:\:\:\:\:\mathrm{Prove}\:\mathrm{that}; \\ $$$$\:\:\:\:\frac{{d}}{{dx}}\:\left(\frac{\mathrm{sin}^{\:\mathrm{2}} {x}}{\mathrm{1}+\mathrm{cot}\:{x}}\:+\:\frac{\mathrm{cos}^{\:\mathrm{2}} {x}}{\mathrm{1}+\mathrm{tan}\:{x}}\right)\:=\:−\mathrm{cos}\:\mathrm{2}{x}\:\:\:\: \\ $$$$ \\ $$
Question Number 219866 Answers: 1 Comments: 0
$$\:\: \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\mathrm{Prove}\:\mathrm{that}; \\ $$$$\:\:\:\:\:\underset{\:\mathrm{0}} {\int}^{\:\pi/\mathrm{2}} \:\mathrm{sin}^{\mathrm{2}{x}−\mathrm{1}} \theta\:\mathrm{cos}\:^{\mathrm{2}{y}−\mathrm{1}} \theta\:{d}\theta\:=\:\frac{\mathrm{1}}{\mathrm{2}}\:\frac{\Gamma\left({x}\right)\Gamma\left({y}\right)}{\Gamma\left({x}\right)+\Gamma\left({y}\right)}\:\:\:\:\: \\ $$$$\: \\ $$
Question Number 219865 Answers: 1 Comments: 0
$$ \\ $$$$\:\:\:\:\:\:\:\:\:\:\mathrm{Prove}\:\mathrm{that}; \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\underset{\:\mathrm{0}} {\overset{\:\mathrm{1}} {\int}}\:{ln}\Gamma\left({x}\right){dx}\:=\:{ln}\:\sqrt{\mathrm{2}\pi} \\ $$$$ \\ $$
Question Number 219864 Answers: 1 Comments: 0
$$ \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\mathrm{Prove}\:\mathrm{that}; \\ $$$$\:\:\:\:\:\:\int_{\:\mathrm{0}} ^{\:\mathrm{1}} \:\frac{{x}^{\:{n}+\mathrm{1}} }{{x}+\mathrm{1}}\:{dx}\:<\:\frac{\mathrm{1}}{\mathrm{2}\left({n}+\mathrm{1}\right)} \\ $$$$ \\ $$
Question Number 219863 Answers: 1 Comments: 0
$$ \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\mathrm{Prove}\:\mathrm{that}; \\ $$$$\underset{\:\mathrm{0}} {\int}^{\:\infty} \:\frac{{ln}\:{x}}{{x}^{\mathrm{3}} +\:{x}\sqrt{{x}}\:+\:\mathrm{1}}\:{dx}\:=\:−\frac{\mathrm{32}\pi}{\mathrm{81}}{sin}\frac{\pi}{\mathrm{18}}\:\:\:\: \\ $$$$ \\ $$
Question Number 219853 Answers: 1 Comments: 0
$$\mathrm{If}\:\:\:\mathrm{0}<\mathrm{a}\leqslant\mathrm{b} \\ $$$$\mathrm{Then}\:\mathrm{prove}\:\mathrm{that} \\ $$$$\int_{\boldsymbol{\mathrm{a}}} ^{\:\boldsymbol{\mathrm{b}}} \left(\mathrm{sinx}\right)^{\mathrm{2}\boldsymbol{\mathrm{sin}}^{\mathrm{2}} \boldsymbol{\mathrm{x}}} \:\centerdot\:\left(\mathrm{1}−\mathrm{sin}^{\mathrm{2}} \mathrm{x}\right)^{\mathrm{1}−\boldsymbol{\mathrm{sin}}^{\mathrm{2}} \boldsymbol{\mathrm{x}}} \:\mathrm{dx}\:\geqslant\:\frac{\mathrm{b}−\mathrm{a}}{\mathrm{2}} \\ $$
Question Number 219849 Answers: 0 Comments: 0
Question Number 219846 Answers: 1 Comments: 0
$$\mathrm{If}\:\:\:\mathrm{f}:\left[\mathrm{a},\mathrm{b}\right]\rightarrow\mathbb{R} \\ $$$$\:\:\:\:\:\:\mathrm{0}<\mathrm{a}\leqslant\mathrm{b} \\ $$$$\mathrm{f}\:-\:\mathrm{continuous} \\ $$$$\mathrm{Then}\:\mathrm{prove}\:\mathrm{that} \\ $$$$\frac{\mathrm{b}^{\mathrm{4047}} \:−\:\mathrm{a}^{\mathrm{4047}} }{\mathrm{4047}}\:+\:\int_{\boldsymbol{\mathrm{a}}} ^{\:\boldsymbol{\mathrm{b}}} \:\mathrm{f}\:^{\mathrm{2}} \:\left(\mathrm{x}^{\mathrm{2024}} \right)\:\mathrm{dx}\:\geqslant\:\frac{\mathrm{1}}{\mathrm{1012}}\:\int_{\boldsymbol{\mathrm{a}}^{\mathrm{2024}} } ^{\:\boldsymbol{\mathrm{b}}^{\mathrm{2024}} } \:\mathrm{f}\left(\mathrm{x}\right)\:\mathrm{dx} \\ $$
Question Number 219844 Answers: 2 Comments: 0
Question Number 219843 Answers: 0 Comments: 0
$$\mathrm{If}\:\:\:\mathrm{a},\mathrm{b},\mathrm{c},\mathrm{d}\:>\:\mathrm{0} \\ $$$$\:\:\:\:\:\:\:\mathrm{a}^{\mathrm{2}} +\mathrm{b}^{\mathrm{2}} +\mathrm{c}^{\mathrm{2}} +\mathrm{d}^{\mathrm{2}} \:=\:\mathrm{4} \\ $$$$\mathrm{Then}\:\mathrm{prove}\:\mathrm{that} \\ $$$$\frac{\mathrm{1}}{\left(\mathrm{1}+\mathrm{ab}\right)^{\mathrm{3}} }\:+\:\frac{\mathrm{1}}{\left(\mathrm{1}+\mathrm{ac}\right)^{\mathrm{3}} }\:+\:\frac{\mathrm{1}}{\left(\mathrm{1}+\mathrm{ad}\right)^{\mathrm{3}} }\:+\:\frac{\mathrm{1}}{\left(\mathrm{1}+\mathrm{bc}\right)^{\mathrm{3}} }\:+\:\frac{\mathrm{1}}{\left(\mathrm{1}+\mathrm{bd}\right)^{\mathrm{3}} }\:+\:\frac{\mathrm{1}}{\left(\mathrm{1}+\mathrm{cd}\right)^{\mathrm{3}} }\:\geqslant\:\frac{\mathrm{3}}{\mathrm{4}} \\ $$
Question Number 219839 Answers: 1 Comments: 0
Question Number 219833 Answers: 0 Comments: 0
Question Number 219832 Answers: 1 Comments: 0
$$ \\ $$$$\:\:\:\:\:{lim}_{{n}\rightarrow\infty} \:{n}\left(\frac{\mathrm{1}}{\mathrm{1}+{n}}\:+\frac{\mathrm{1}}{\mathrm{2}+{n}}\:+...+\frac{\mathrm{1}}{\mathrm{2}{n}}\:−{ln}\left(\mathrm{2}\right)\right)=? \\ $$$$ \\ $$
Question Number 219831 Answers: 1 Comments: 0
$$\mathrm{prove} \\ $$$$\underset{{h}\rightarrow\mathrm{0}} {\mathrm{lim}}\left(\frac{\mathrm{g}\left({z}+{h}\right)}{\mathrm{g}\left({z}\right)}\right)^{\frac{\mathrm{1}}{{h}}} ={e}^{\frac{\mathrm{d}\:\:}{\mathrm{d}{z}}\:\mathrm{ln}\:\left(\mathrm{g}\left({z}\right)\right)} ={e}^{\frac{\mathrm{g}^{\left(\mathrm{1}\right)} \left({z}\right)}{\mathrm{g}\left({z}\right)}} \\ $$
Question Number 219806 Answers: 2 Comments: 0
$$\underset{{h}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\:\left(\frac{\mathrm{cos}\left({x}+{h}\right)}{\mathrm{cos}\left({x}\right)}\right)^{\frac{\mathrm{1}}{{h}}} =?? \\ $$
Question Number 219801 Answers: 1 Comments: 0
$$\underset{{h}\rightarrow\infty} {\mathrm{lim}}\:{h}^{\nu} {J}_{\nu} \left({h}\right)=?? \\ $$
Question Number 219800 Answers: 0 Comments: 0
$${y}^{\left(\mathrm{2}\right)} \left({t}\right)=\left(\mathrm{1}−{e}^{{t}} \right){y}\left({t}\right)+{y}^{\left(\mathrm{1}\right)} \left({t}\right) \\ $$
Question Number 219799 Answers: 0 Comments: 0
$${y}^{\left(\mathrm{2}\right)} \left({t}\right)+{y}\left({t}\right)=\mathrm{cos}\left({t}\right) \\ $$
Question Number 219798 Answers: 0 Comments: 0
$$\mathrm{solve}\: \\ $$$${y}^{\left(\mathrm{2}\right)} \left({t}\right)={y}^{\left(\mathrm{1}\right)} \left({t}\right){e}^{−{y}\left({t}\right)} \\ $$
Question Number 219797 Answers: 0 Comments: 0
$$\mathrm{solve} \\ $$$$\left({y}^{\left(\mathrm{2}\right)} \left({t}\right)\right)^{\mathrm{2}} =\frac{\mathrm{1}}{\mathrm{1}+{y}\left({t}\right)} \\ $$
Question Number 219796 Answers: 0 Comments: 0
$$\mathrm{Solve} \\ $$$${y}^{\left(\mathrm{2}\right)} \left({t}\right)=\left({y}\left({t}\right)\right)^{\mathrm{2}} −{ay}^{\left(\mathrm{1}\right)} \left({t}\right)−{by}\left({t}\right) \\ $$
Question Number 219795 Answers: 1 Comments: 0
$$\int_{\mathcal{D}=\left[\mathrm{0},\mathrm{1}\right]^{{N}} } \:\underset{{h}=\mathrm{1}} {\overset{{N}} {\prod}}\:{e}^{−\frac{\mathrm{1}}{\mathrm{2}}{x}_{{h}} } \mathrm{d}{x}_{{h}} \\ $$
Question Number 219794 Answers: 0 Comments: 0
$$\int_{\mathrm{0}} ^{\:\infty} \:\:\frac{{Y}_{\mathrm{0}} \left({t}\right){e}^{−\mathrm{3}{t}} }{{t}^{\mathrm{2}} +\mathrm{1}}\:\mathrm{d}{t} \\ $$
Question Number 219793 Answers: 1 Comments: 0
$$\int_{\mathrm{0}} ^{\:\infty} \:\:\frac{{e}^{−{t}} }{{t}^{\mathrm{2}} +\mathrm{1}}\:\mathrm{d}{t}=? \\ $$
Question Number 219792 Answers: 1 Comments: 0
$$\underset{{z}\rightarrow\infty} {\mathrm{lim}}\:\frac{{J}_{\mathrm{1}} \left({z}\right)}{{Y}_{\mathrm{0}} \left({z}\right)} \\ $$$$\underset{{z}\rightarrow\mathrm{0}} {\mathrm{lim}}\:{z}\frac{\mathrm{1}−\frac{\mathrm{1}}{\mathrm{2}}{z}^{\mathrm{2}} −\mathrm{cos}\left(\frac{{z}}{\mathrm{1}−{z}^{\mathrm{2}} }\right)}{{z}^{\mathrm{4}} } \\ $$$$\underset{{z}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\frac{{J}_{\nu} \left({z}+{h}\right){Y}_{\nu} \left({z}\right)−{J}_{\nu} \left({z}\right){Y}_{\nu} \left({z}\right)}{{h}}\: \\ $$
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