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Question Number 164124    Answers: 1   Comments: 0

Solve for real numbers: ((16^x + 4^x + 1^x )/(4^x + 2^x + 1^x )) = ((8^x + 1)/(65))

$$\mathrm{Solve}\:\mathrm{for}\:\mathrm{real}\:\mathrm{numbers}: \\ $$$$\frac{\mathrm{16}^{\boldsymbol{\mathrm{x}}} \:+\:\mathrm{4}^{\boldsymbol{\mathrm{x}}} \:+\:\mathrm{1}^{\boldsymbol{\mathrm{x}}} }{\mathrm{4}^{\boldsymbol{\mathrm{x}}} \:+\:\mathrm{2}^{\boldsymbol{\mathrm{x}}} \:+\:\mathrm{1}^{\boldsymbol{\mathrm{x}}} }\:=\:\frac{\mathrm{8}^{\boldsymbol{\mathrm{x}}} \:+\:\mathrm{1}}{\mathrm{65}} \\ $$

Question Number 164123    Answers: 0   Comments: 0

very nice to problem: find in closed form; ∫_0 ^1 log (1−x^2 ) log^(n ) (1−x) dx; n ∈ N^+ ^(z.)

$$\mathrm{very}\:\mathrm{nice}\:\mathrm{to}\:\mathrm{problem}: \\ $$$$\boldsymbol{{find}}\:\boldsymbol{{in}}\:\boldsymbol{{closed}}\:\boldsymbol{{form}}; \\ $$$$\:\int_{\mathrm{0}} ^{\mathrm{1}} \:\boldsymbol{{log}}\:\left(\mathrm{1}−\boldsymbol{{x}}^{\mathrm{2}} \right)\:\boldsymbol{{log}}\:^{\boldsymbol{{n}}\:} \:\left(\mathrm{1}−\boldsymbol{{x}}\right)\:\boldsymbol{{dx}}; \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\boldsymbol{{n}}\:\in\:\:\mathbb{N}^{+} \\ $$$$\:^{\mathrm{z}.} \\ $$

Question Number 164122    Answers: 1   Comments: 0

lim_(n→∞) Σ_(k=1) ^n sin (1/(n+k))=?

$$\underset{\mathrm{n}\rightarrow\infty} {\mathrm{lim}}\underset{\mathrm{k}=\mathrm{1}} {\overset{\mathrm{n}} {\sum}}\mathrm{sin}\:\frac{\mathrm{1}}{\mathrm{n}+\mathrm{k}}=? \\ $$

Question Number 164121    Answers: 1   Comments: 3

if w = f(x,y) and x = r cosθ , y = rsinθ then prove that w_(rr) + w_(θθ) = 0?

$${if}\:{w}\:=\:{f}\left({x},{y}\right)\:{and}\:{x}\:=\:{r}\:{cos}\theta\:,\:{y}\:=\:{rsin}\theta \\ $$$$ \\ $$$${then}\:{prove}\:{that}\:{w}_{{rr}} \:+\:{w}_{\theta\theta} \:=\:\mathrm{0}? \\ $$

Question Number 164120    Answers: 0   Comments: 1

How do you all to prove Integral; Prove the; ∫ (((In x)2)/x) dx = (1/3) (In x)^3

$$\mathrm{How}\:\mathrm{do}\:\mathrm{you}\:\mathrm{all}\:\mathrm{to}\:\mathrm{prove}\:\mathrm{Integral}; \\ $$$$\:\boldsymbol{{Prove}}\:\boldsymbol{{the}}; \\ $$$$\:\int\:\frac{\left(\boldsymbol{{In}}\:\boldsymbol{{x}}\right)\mathrm{2}}{\boldsymbol{{x}}}\:\boldsymbol{{dx}}\:=\:\frac{\mathrm{1}}{\mathrm{3}}\:\left(\boldsymbol{{In}}\:\boldsymbol{{x}}\right)^{\mathrm{3}} \\ $$

Question Number 164117    Answers: 0   Comments: 0

soit (a_n )_n une suite define par a_n =Σ_(k=1) ^n (1/k)−ln(n) montrons a_n converge

$${soit}\:\left({a}_{{n}} \right)_{{n}} {une}\:{suite}\:{define}\:{par} \\ $$$${a}_{{n}} =\underset{{k}=\mathrm{1}} {\overset{{n}} {\sum}}\frac{\mathrm{1}}{{k}}−{ln}\left({n}\right)\:\:{montrons}\: \\ $$$${a}_{{n}} {converge} \\ $$

Question Number 164116    Answers: 0   Comments: 1

(a+b+c+x)^(100) 50^(th) limit is equl to=?

$$\left({a}+{b}+{c}+{x}\right)^{\mathrm{100}} \\ $$$$\mathrm{50}^{{th}} \:{limit}\:{is}\:{equl}\:{to}=? \\ $$

Question Number 164118    Answers: 1   Comments: 0

What is Mathematical Calculus and Mathematical fhysics, Then what is the difference and how to explain?? ^([Zaynal])

$$\boldsymbol{{What}}\:\boldsymbol{{is}} \\ $$$$\:\boldsymbol{{Mathematical}}\:\boldsymbol{{Calculus}}\:\boldsymbol{{and}} \\ $$$$\boldsymbol{{Mathematical}}\:\boldsymbol{{fhysics}}, \\ $$$$\boldsymbol{{Then}}\:\boldsymbol{{what}}\:\boldsymbol{{is}}\:\boldsymbol{{the}}\:\boldsymbol{{difference}}\:\boldsymbol{{and}} \\ $$$$\boldsymbol{{how}}\:\boldsymbol{{to}}\:\boldsymbol{{explain}}?? \\ $$$$\:^{\left[\mathrm{Zaynal}\right]} \\ $$

Question Number 164103    Answers: 1   Comments: 0

prove that Ω=∫_0 ^( 1) ln(((1+x)/(1−x)) ).(dx/(x (√( 1−x^( 2) )))) = (π^( 2) /2) −− m.n−−

$$ \\ $$$$\:\:\:{prove}\:{that} \\ $$$$\: \\ $$$$\:\:\:\Omega=\int_{\mathrm{0}} ^{\:\mathrm{1}} \mathrm{ln}\left(\frac{\mathrm{1}+{x}}{\mathrm{1}−{x}}\:\right).\frac{{dx}}{{x}\:\sqrt{\:\mathrm{1}−{x}^{\:\mathrm{2}} }}\:=\:\frac{\pi^{\:\mathrm{2}} }{\mathrm{2}} \\ $$$$\:\:\:\:\:−−\:{m}.{n}−− \\ $$$$ \\ $$

Question Number 164468    Answers: 2   Comments: 0

Question Number 164466    Answers: 0   Comments: 1

Question Number 164085    Answers: 0   Comments: 0

Question Number 164077    Answers: 1   Comments: 0

Question Number 164073    Answers: 2   Comments: 1

Question Number 164071    Answers: 1   Comments: 0

Question Number 164068    Answers: 1   Comments: 0

Question Number 164059    Answers: 0   Comments: 0

Air leaks from a spherical ballon so that it maintains its shape at a rate of 25 cc/m .What is the rate of change in the length of the radius of the balloon when the radius is 5 cm

$$\:\:\mathrm{Air}\:\mathrm{leaks}\:\mathrm{from}\:\mathrm{a}\:\mathrm{spherical}\:\mathrm{ballon}\:\mathrm{so}\:\mathrm{that}\: \\ $$$$\:\mathrm{it}\:\mathrm{maintains}\:\mathrm{its}\:\mathrm{shape}\:\mathrm{at}\:\mathrm{a}\:\mathrm{rate}\:\mathrm{of}\:\mathrm{25}\:\mathrm{cc}/\mathrm{m} \\ $$$$\:.\mathrm{What}\:\mathrm{is}\:\mathrm{the}\:\mathrm{rate}\:\mathrm{of}\:\mathrm{change}\:\mathrm{in}\:\mathrm{the}\:\mathrm{length} \\ $$$$\:\:\mathrm{of}\:\mathrm{the}\:\mathrm{radius}\:\mathrm{of}\:\mathrm{the}\:\mathrm{balloon}\:\mathrm{when}\:\mathrm{the}\:\mathrm{radius} \\ $$$$\:\:\mathrm{is}\:\mathrm{5}\:\mathrm{cm} \\ $$

Question Number 164052    Answers: 0   Comments: 0

Question Number 164050    Answers: 1   Comments: 0

Question Number 164039    Answers: 1   Comments: 0

consider f function Df=[0,1] f(0)=f(1) c∈[0,(1/2)] show that f(c)=f(c+(1/2))

$${consider}\:{f}\:{function}\:{Df}=\left[\mathrm{0},\mathrm{1}\right] \\ $$$${f}\left(\mathrm{0}\right)={f}\left(\mathrm{1}\right)\:{c}\in\left[\mathrm{0},\frac{\mathrm{1}}{\mathrm{2}}\right]\:{show}\:{that}\:{f}\left({c}\right)={f}\left({c}+\frac{\mathrm{1}}{\mathrm{2}}\right) \\ $$

Question Number 164027    Answers: 1   Comments: 2

Question Number 164024    Answers: 1   Comments: 0

Question Number 164019    Answers: 0   Comments: 0

x^4 +ax^3 +bx^2 +cx+d=0 ⇒ x^2 +(d/x^2 )+ax+(c/x)+b=0 say (1/x)=t ⇒ x^2 +ax+b+dt^2 +ct=0 let x^2 +ax+p=0 & dt^2 +ct+q=0 p+q=b x=−(a/2)±(√((a^2 /4)−p)) t=−(c/(2d))±(√((c^2 /(4d^2 ))−(q/d))) tx=1 ⇒ ((ac)/(4d))∓(a/2)(√((c^2 /(4d^2 ))−(q/d)))∓(c/(2d))(√((a^2 /4)−p)) +(√((a^2 /4)−p))(√((c^2 /(4d^2 ))−(q/d))) =1 let p=(a^2 /4) ⇒ ((ac)/(4d))∓(a/2)(√((c^2 /(4d^2 ))−(b/d)+(a^2 /(4d))))=1 ⇒ (c^2 /(4d^2 ))−(b/d)+(a^2 /(4d))=(4/a^2 )+(c^2 /(4d^2 ))−((2c)/(ad)) ⇒ 4a^2 b−a^4 +16d−8ac=0 if x=s+h s^4 +4hs(s^2 +h^2 )+6h^2 s^2 +h^4 +a{s^3 +3hs(s+h)+h^2 } +b{s^2 +2hs+h^2 }+c(s+h)+d=0 A=4h+a B=6h^2 +3ah+b C=4h^3 +3ah^2 +2bh+c D=h^4 +ah^3 +bh^2 +ch+d ⇒ (4h+a)^2 (8h^2 +4ah+4b−a^2 ) +16(h^4 +ah^3 +bh^2 +ch+d) =8(4h+a)(4h^3 +3ah^2 +2bh+c) .....

$$\:\:{x}^{\mathrm{4}} +{ax}^{\mathrm{3}} +{bx}^{\mathrm{2}} +{cx}+{d}=\mathrm{0} \\ $$$$\:\Rightarrow\:\:{x}^{\mathrm{2}} +\frac{{d}}{{x}^{\mathrm{2}} }+{ax}+\frac{{c}}{{x}}+{b}=\mathrm{0} \\ $$$${say}\:\:\:\frac{\mathrm{1}}{{x}}={t}\:\:\Rightarrow \\ $$$$\:\:{x}^{\mathrm{2}} +{ax}+{b}+{dt}^{\mathrm{2}} +{ct}=\mathrm{0} \\ $$$${let}\:\:\:\:{x}^{\mathrm{2}} +{ax}+{p}=\mathrm{0} \\ $$$$\&\:\:\:\:\:\:{dt}^{\mathrm{2}} +{ct}+{q}=\mathrm{0} \\ $$$$\:\:\:\:{p}+{q}={b} \\ $$$${x}=−\frac{{a}}{\mathrm{2}}\pm\sqrt{\frac{{a}^{\mathrm{2}} }{\mathrm{4}}−{p}}\:\: \\ $$$${t}=−\frac{{c}}{\mathrm{2}{d}}\pm\sqrt{\frac{{c}^{\mathrm{2}} }{\mathrm{4}{d}^{\mathrm{2}} }−\frac{{q}}{{d}}} \\ $$$${tx}=\mathrm{1}\:\:\:\:\Rightarrow \\ $$$$\frac{{ac}}{\mathrm{4}{d}}\mp\frac{{a}}{\mathrm{2}}\sqrt{\frac{{c}^{\mathrm{2}} }{\mathrm{4}{d}^{\mathrm{2}} }−\frac{{q}}{{d}}}\mp\frac{{c}}{\mathrm{2}{d}}\sqrt{\frac{{a}^{\mathrm{2}} }{\mathrm{4}}−{p}} \\ $$$$\:\:\:\:\:\:\:\:+\sqrt{\frac{{a}^{\mathrm{2}} }{\mathrm{4}}−{p}}\sqrt{\frac{{c}^{\mathrm{2}} }{\mathrm{4}{d}^{\mathrm{2}} }−\frac{{q}}{{d}}}\:=\mathrm{1} \\ $$$${let}\:\:\:{p}=\frac{{a}^{\mathrm{2}} }{\mathrm{4}}\:\:\Rightarrow \\ $$$$\:\:\:\frac{{ac}}{\mathrm{4}{d}}\mp\frac{{a}}{\mathrm{2}}\sqrt{\frac{{c}^{\mathrm{2}} }{\mathrm{4}{d}^{\mathrm{2}} }−\frac{{b}}{{d}}+\frac{{a}^{\mathrm{2}} }{\mathrm{4}{d}}}=\mathrm{1} \\ $$$$\Rightarrow\:\:\cancel{\frac{{c}^{\mathrm{2}} }{\mathrm{4}{d}^{\mathrm{2}} }}−\frac{{b}}{{d}}+\frac{{a}^{\mathrm{2}} }{\mathrm{4}{d}}=\frac{\mathrm{4}}{{a}^{\mathrm{2}} }+\cancel{\frac{{c}^{\mathrm{2}} }{\mathrm{4}{d}^{\mathrm{2}} }}−\frac{\mathrm{2}{c}}{{ad}} \\ $$$$\Rightarrow\:\:\:\mathrm{4}{a}^{\mathrm{2}} {b}−{a}^{\mathrm{4}} +\mathrm{16}{d}−\mathrm{8}{ac}=\mathrm{0} \\ $$$${if}\:{x}={s}+{h} \\ $$$${s}^{\mathrm{4}} +\mathrm{4}{hs}\left({s}^{\mathrm{2}} +{h}^{\mathrm{2}} \right)+\mathrm{6}{h}^{\mathrm{2}} {s}^{\mathrm{2}} +{h}^{\mathrm{4}} \\ $$$$+{a}\left\{{s}^{\mathrm{3}} +\mathrm{3}{hs}\left({s}+{h}\right)+{h}^{\mathrm{2}} \right\} \\ $$$$\:\:+{b}\left\{{s}^{\mathrm{2}} +\mathrm{2}{hs}+{h}^{\mathrm{2}} \right\}+{c}\left({s}+{h}\right)+{d}=\mathrm{0} \\ $$$${A}=\mathrm{4}{h}+{a} \\ $$$${B}=\mathrm{6}{h}^{\mathrm{2}} +\mathrm{3}{ah}+{b} \\ $$$${C}=\mathrm{4}{h}^{\mathrm{3}} +\mathrm{3}{ah}^{\mathrm{2}} +\mathrm{2}{bh}+{c} \\ $$$${D}={h}^{\mathrm{4}} +{ah}^{\mathrm{3}} +{bh}^{\mathrm{2}} +{ch}+{d} \\ $$$$\Rightarrow \\ $$$$\left(\mathrm{4}{h}+{a}\right)^{\mathrm{2}} \left(\mathrm{8}{h}^{\mathrm{2}} +\mathrm{4}{ah}+\mathrm{4}{b}−{a}^{\mathrm{2}} \right) \\ $$$$+\mathrm{16}\left({h}^{\mathrm{4}} +{ah}^{\mathrm{3}} +{bh}^{\mathrm{2}} +{ch}+{d}\right) \\ $$$$\:\:\:\:\:\:=\mathrm{8}\left(\mathrm{4}{h}+{a}\right)\left(\mathrm{4}{h}^{\mathrm{3}} +\mathrm{3}{ah}^{\mathrm{2}} +\mathrm{2}{bh}+{c}\right) \\ $$$$..... \\ $$$$\:\:\:\:\:\: \\ $$

Question Number 164018    Answers: 1   Comments: 0

Question Number 164028    Answers: 1   Comments: 0

Question Number 164010    Answers: 0   Comments: 3

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