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Question Number 220262    Answers: 7   Comments: 0

Question Number 220257    Answers: 2   Comments: 0

proof that volume of frustum of circular cone is (1/3)h[A1+A2+(√(A1A2)) A_1 and A_2 are areas of base

$${proof}\:{that}\:{volume}\:{of}\:{frustum}\:{of} \\ $$$$\:{circular}\:{cone}\:{is}\:\frac{\mathrm{1}}{\mathrm{3}}{h}\left[{A}\mathrm{1}+{A}\mathrm{2}+\sqrt{{A}\mathrm{1}{A}\mathrm{2}}\right. \\ $$$${A}_{\mathrm{1}} {and}\:{A}_{\mathrm{2}} \:{are}\:\:{areas}\:{of}\:{base} \\ $$

Question Number 220253    Answers: 0   Comments: 0

Question Number 220250    Answers: 3   Comments: 0

Question Number 220249    Answers: 1   Comments: 0

Question Number 220248    Answers: 20   Comments: 0

Question Number 220247    Answers: 2   Comments: 0

Question Number 220246    Answers: 6   Comments: 0

Question Number 220245    Answers: 1   Comments: 0

Question Number 220244    Answers: 3   Comments: 0

Question Number 220243    Answers: 5   Comments: 0

Question Number 220242    Answers: 0   Comments: 0

∫ ((ln x)/((1 + x^2 )^2 )) dx

$$ \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\int\:\:\frac{{ln}\:{x}}{\left(\mathrm{1}\:+\:{x}^{\mathrm{2}} \right)\:^{\mathrm{2}} }\:\:{dx} \\ $$$$ \\ $$

Question Number 220232    Answers: 1   Comments: 0

prove that (π/(16)) < ∫_0 ^( 1 ) (√((x(1−x))/(sin(πx)+cos(πx)+2))) dx<(π/8)

$$ \\ $$$$\:\:\:\:{prove}\:{that} \\ $$$$ \\ $$$$\:\:\frac{\pi}{\mathrm{16}}\:<\:\int_{\mathrm{0}} ^{\:\mathrm{1}\:} \sqrt{\frac{{x}\left(\mathrm{1}−{x}\right)}{{sin}\left(\pi{x}\right)+{cos}\left(\pi{x}\right)+\mathrm{2}}}\:{dx}<\frac{\pi}{\mathrm{8}} \\ $$$$\:\:\:\:\: \\ $$

Question Number 220231    Answers: 4   Comments: 0

Question Number 220224    Answers: 5   Comments: 0

calcul together of definition of and calcul the derive f^ ^′ f(x)= x(√((x−1)/(x+1)))

$${calcul}\:{together}\:{of}\:{definition}\:{of}\:{and} \\ $$$${calcul}\:{the}\:{derive}\:{f}^{\:} \:^{'} \\ $$$${f}\left({x}\right)=\:\:{x}\sqrt{\frac{{x}−\mathrm{1}}{{x}+\mathrm{1}}} \\ $$

Question Number 220221    Answers: 0   Comments: 2

Q.The density of an object of mass M is δ and the density of the air is ρ. the mass of of the object is measured with the help of a metal weight of mass m . the density of the metal weight is d. if ρ≪δ them show that the real mass M will be m(1−(ρ/d) )(1+(ρ/δ)) I have managed to M=((m(1−(ρ/d)))/((1−(ρ/δ)))) but I can not figure it to the end please help

$${Q}.{The}\:{density}\:{of}\:{an}\:{object}\:{of}\:{mass}\:{M}\:{is}\:\delta\:{and}\:{the}\:{density}\:{of}\:{the}\:{air}\:{is}\:\rho. \\ $$$${the}\:{mass}\:{of}\:{of}\:{the}\:{object}\:{is}\:{measured}\:{with}\:\:{the}\:{help}\:{of}\:{a}\:{metal}\:{weight}\:{of}\:{mass}\:{m}\:. \\ $$$${the}\:{density}\:{of}\:{the}\:{metal}\:{weight}\:{is}\:{d}. \\ $$$${if}\:\rho\ll\delta\:{them}\:{show}\:{that}\:{the}\:{real}\:{mass}\:{M}\:{will}\:{be} \\ $$$${m}\left(\mathrm{1}−\frac{\rho}{{d}}\:\right)\left(\mathrm{1}+\frac{\rho}{\delta}\right) \\ $$$${I}\:{have}\:{managed}\:{to}\:{M}=\frac{{m}\left(\mathrm{1}−\frac{\rho}{{d}}\right)}{\left(\mathrm{1}−\frac{\rho}{\delta}\right)} \\ $$$${but}\:{I}\:{can}\:{not}\:{figure}\:{it}\:{to}\:{the}\:{end} \\ $$$${please}\:{help} \\ $$

Question Number 220208    Answers: 3   Comments: 2

Question Number 220200    Answers: 0   Comments: 0

∫_1 ^( α) (((x − 1)^n )/(e^x − x − 1))dx

$$ \\ $$$$\int_{\mathrm{1}} ^{\:\alpha} \frac{\left({x}\:−\:\mathrm{1}\right)^{{n}} }{{e}^{{x}} \:−\:{x}\:−\:\mathrm{1}}{dx} \\ $$

Question Number 220193    Answers: 3   Comments: 0

x^8 =21x+13 ; x∈R x=?

$$\:\:\:\:\:\:\:\:\boldsymbol{{x}}^{\mathrm{8}} =\mathrm{21}\boldsymbol{{x}}+\mathrm{13}\:\:\:\:\:\:\:\:\:\:;\:\:\:\:\boldsymbol{{x}}\in{R} \\ $$$$\:\:\:\:\boldsymbol{{x}}=? \\ $$

Question Number 220192    Answers: 0   Comments: 0

Question Number 220190    Answers: 1   Comments: 0

i^i =e^(−(π/2)) and we can renote complex number i as ((0,(−1)),(1,( 0)) ) i^i = ((0,(−1)),(1,( 0)) )^ ((0,(−1)),(1,( 0)) ) But why Matrix Exponent Calculate Dosen′t defined?? I mean A,B∈mat(m,m) why A^B dosen′t defined??

$${i}^{{i}} ={e}^{−\frac{\pi}{\mathrm{2}}} \: \\ $$$$\mathrm{and}\:\mathrm{we}\:\mathrm{can}\:\mathrm{renote}\:\mathrm{complex}\:\mathrm{number}\:\boldsymbol{{i}}\:\mathrm{as}\:\begin{pmatrix}{\mathrm{0}}&{−\mathrm{1}}\\{\mathrm{1}}&{\:\:\:\:\mathrm{0}}\end{pmatrix} \\ $$$$\boldsymbol{{i}}^{\boldsymbol{{i}}} =\begin{pmatrix}{\mathrm{0}}&{−\mathrm{1}}\\{\mathrm{1}}&{\:\:\:\:\mathrm{0}}\end{pmatrix}^{\begin{pmatrix}{\mathrm{0}}&{−\mathrm{1}}\\{\mathrm{1}}&{\:\:\:\:\mathrm{0}}\end{pmatrix}} \: \\ $$$$\:\mathrm{But}\:\mathrm{why}\:\mathrm{Matrix}\:\mathrm{Exponent}\:\mathrm{Calculate}\:\mathrm{Dosen}'\mathrm{t}\:\mathrm{defined}?? \\ $$$$\:\mathrm{I}\:\mathrm{mean}\:{A},{B}\in\mathrm{mat}\left({m},{m}\right) \\ $$$$\mathrm{why}\:\mathrm{A}^{\mathrm{B}} \:\mathrm{dosen}'\mathrm{t}\:\mathrm{defined}?? \\ $$

Question Number 220184    Answers: 2   Comments: 0

∫_0 ^(π/(12)) (√((sec^4 α+5sec^5 αsin α)/((2−sec^2 α)(125tan^3 α+25tan^2 α+5tan α+1))))dα

$$\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{12}}} \sqrt{\frac{\mathrm{sec}^{\mathrm{4}} \alpha+\mathrm{5sec}^{\mathrm{5}} \alpha\mathrm{sin}\:\alpha}{\left(\mathrm{2}−\mathrm{sec}^{\mathrm{2}} \alpha\right)\left(\mathrm{125tan}^{\mathrm{3}} \alpha+\mathrm{25tan}^{\mathrm{2}} \alpha+\mathrm{5tan}\:\alpha+\mathrm{1}\right)}}{d}\alpha \\ $$

Question Number 220323    Answers: 0   Comments: 0

Q1. 𝛀;={(x,y);x^2 +y^2 ≤1} (1/2) ∮_( ∂𝛀) x∙dy−y∙dy=?? Q2. S; R^2 →R^3 S(u,v)=rsin(u)cos(v)e_1 ^→ +rsin(u)sin(v)e_2 ^→ +rcos(u)e_3 ^→ F^→ ;R^3 →R^3 F^→ (x,y,z)=−(x/( (√(x^2 +y^2 +z^2 ))))e_1 ^→ −(y/( (√(x^2 +y^2 +z^2 ))))e_2 ^→ −(z/( (√(x^2 +y^2 +z^2 ))))e_3 ^→ ∫∫_( D) F^→ ∙dS^→ =∫∫∫_( K) ▽^→ ∙F^→ dV=??? Q3. if ∮_( C) F^→ ∙dl=0 Prove ▽^→ ×F^→ =0 Q4. Prove if ▽^→ ×F^→ ≠0 → ∮_( C) F^→ ∙dl≠0

$$\mathrm{Q1}.\:\boldsymbol{\Omega};=\left\{\left({x},{y}\right);{x}^{\mathrm{2}} +{y}^{\mathrm{2}} \leq\mathrm{1}\right\} \\ $$$$\frac{\mathrm{1}}{\mathrm{2}}\:\oint_{\:\partial\boldsymbol{\Omega}} \:{x}\centerdot\mathrm{d}{y}−{y}\centerdot\mathrm{d}{y}=?? \\ $$$$\mathrm{Q2}.\:\boldsymbol{\mathcal{S}};\:\mathbb{R}^{\mathrm{2}} \rightarrow\mathbb{R}^{\mathrm{3}} \\ $$$$\boldsymbol{\mathcal{S}}\left({u},{v}\right)={r}\mathrm{sin}\left({u}\right)\mathrm{cos}\left({v}\right)\overset{\rightarrow} {\boldsymbol{\mathrm{e}}}_{\mathrm{1}} +{r}\mathrm{sin}\left({u}\right)\mathrm{sin}\left({v}\right)\overset{\rightarrow} {\boldsymbol{\mathrm{e}}}_{\mathrm{2}} +{r}\mathrm{cos}\left({u}\right)\overset{\rightarrow} {\boldsymbol{\mathrm{e}}}_{\mathrm{3}} \\ $$$$\overset{\rightarrow} {\boldsymbol{\mathrm{F}}};\mathbb{R}^{\mathrm{3}} \rightarrow\mathbb{R}^{\mathrm{3}} \\ $$$$\overset{\rightarrow} {\boldsymbol{\mathrm{F}}}\left({x},{y},{z}\right)=−\frac{{x}}{\:\sqrt{{x}^{\mathrm{2}} +{y}^{\mathrm{2}} +{z}^{\mathrm{2}} }}\overset{\rightarrow} {\boldsymbol{\mathrm{e}}}_{\mathrm{1}} −\frac{{y}}{\:\sqrt{{x}^{\mathrm{2}} +{y}^{\mathrm{2}} +{z}^{\mathrm{2}} }}\overset{\rightarrow} {\boldsymbol{\mathrm{e}}}_{\mathrm{2}} −\frac{{z}}{\:\sqrt{{x}^{\mathrm{2}} +{y}^{\mathrm{2}} +{z}^{\mathrm{2}} }}\overset{\rightarrow} {\boldsymbol{\mathrm{e}}}_{\mathrm{3}} \\ $$$$\:\int\int_{\:\mathcal{D}} \:\overset{\rightarrow} {\boldsymbol{\mathrm{F}}}\centerdot\mathrm{d}\overset{\rightarrow} {\boldsymbol{\mathcal{S}}}=\int\int\int_{\:\boldsymbol{{K}}} \:\overset{\rightarrow} {\bigtriangledown}\centerdot\overset{\rightarrow} {\boldsymbol{\mathrm{F}}}\:\mathrm{d}{V}=??? \\ $$$$\mathrm{Q3}. \\ $$$$\mathrm{if}\:\:\oint_{\:\mathcal{C}} \:\overset{\rightarrow} {\boldsymbol{\mathrm{F}}}\centerdot\mathrm{d}\boldsymbol{\mathrm{l}}=\mathrm{0}\:\:\mathrm{Prove}\:\overset{\rightarrow} {\bigtriangledown}×\overset{\rightarrow} {\boldsymbol{\mathrm{F}}}=\mathrm{0} \\ $$$$\mathrm{Q4}. \\ $$$$\mathrm{Prove}\:\:\mathrm{if}\:\overset{\rightarrow} {\bigtriangledown}×\overset{\rightarrow} {\boldsymbol{\mathrm{F}}}\neq\mathrm{0}\:\rightarrow\:\oint_{\:\mathcal{C}} \:\overset{\rightarrow} {\boldsymbol{\mathrm{F}}}\centerdot\mathrm{d}\boldsymbol{\mathrm{l}}\neq\mathrm{0} \\ $$

Question Number 220179    Answers: 1   Comments: 0

∫ (ds/( (√(s^2 +1))(s+(√(s^2 +1)))^(−ν) ))

$$\int\:\:\:\frac{\mathrm{d}{s}}{\:\sqrt{{s}^{\mathrm{2}} +\mathrm{1}}\left({s}+\sqrt{{s}^{\mathrm{2}} +\mathrm{1}}\right)^{−\nu} } \\ $$

Question Number 220178    Answers: 1   Comments: 0

evaluate ∫_w ^( ∞) ((e^s ∙𝚪(0,s))/s)ds

$$\mathrm{evaluate} \\ $$$$\int_{{w}} ^{\:\infty} \:\frac{{e}^{{s}} \centerdot\boldsymbol{\Gamma}\left(\mathrm{0},{s}\right)}{{s}}\mathrm{d}{s} \\ $$

Question Number 220177    Answers: 1   Comments: 0

∫_(−1) ^( 0) cos(((ln(z+1))/z)) dz

$$\int_{−\mathrm{1}} ^{\:\mathrm{0}} \:\:\mathrm{cos}\left(\frac{\mathrm{ln}\left({z}+\mathrm{1}\right)}{{z}}\right)\:\mathrm{d}{z} \\ $$

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