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Question Number 78256 Answers: 0 Comments: 0

Evaluate Σa_1 a_2 a_3 as a function of a_i

$$\mathrm{Evaluate}\:\Sigma\mathrm{a}_{\mathrm{1}} \mathrm{a}_{\mathrm{2}} \mathrm{a}_{\mathrm{3}} \: \\ $$$$\mathrm{as}\:\mathrm{a}\:\mathrm{function}\:\mathrm{of}\:\:\mathrm{a}_{\mathrm{i}} \: \\ $$$$ \\ $$$$ \\ $$

Question Number 78261 Answers: 0 Comments: 3

let U_n =∫_0 ^1 (x^n /(1+x))dx calculate U_n +U_(n+1)

$${let}\:{U}_{{n}} =\int_{\mathrm{0}} ^{\mathrm{1}} \:\frac{{x}^{{n}} }{\mathrm{1}+{x}}{dx}\:\:{calculate} \\ $$$${U}_{{n}} \:+{U}_{{n}+\mathrm{1}} \\ $$

Question Number 78251 Answers: 1 Comments: 0

∫((x+4)/(x−(x)^(1/3) )) dx

$$\int\frac{{x}+\mathrm{4}}{{x}−\sqrt[{\mathrm{3}}]{{x}}}\:{dx}\: \\ $$

Question Number 78246 Answers: 0 Comments: 0

common equation of conic sections ax^2 +bxy+cy^2 +dx+ey+f=0 if b≠0 we rotate tan 2α =(b/(a−c)) [if a=c ⇒ α=45°] { ((x=x′cos α −y′sin α)),((y=x′sin α +y′cos α)) :} we now have [using x, y again instead of x′, y′] Ax^2 +Cy^2 +Dx+Ey+F=0 now complete the squares A(x+(D/(2A)))^2 +C(y+(E/(2C)))^2 +(F−(D^2 /(4A^2 ))−(E^2 /(4C^2 )))=0 { ((x=x′−(D/(2A)))),((y=y′−(E/(2C)))) :} we now have [using x, y again instead of x′, y′] one of these { ((Ax^2 +By^2 +C=0)),((Ax+By^2 +C=0)),((Ax^2 +By+C=0)) :} a, c, A, C, A, B, C ≠0 in all other cases use your brain to interprete which kind of curve (if any) we have

$$\mathrm{common}\:\mathrm{equation}\:\mathrm{of}\:\mathrm{conic}\:\mathrm{sections} \\ $$$${ax}^{\mathrm{2}} +{bxy}+{cy}^{\mathrm{2}} +{dx}+{ey}+{f}=\mathrm{0} \\ $$$$\mathrm{if}\:{b}\neq\mathrm{0}\:\mathrm{we}\:\mathrm{rotate} \\ $$$$\mathrm{tan}\:\mathrm{2}\alpha\:=\frac{{b}}{{a}−{c}}\:\left[\mathrm{if}\:{a}={c}\:\Rightarrow\:\alpha=\mathrm{45}°\right] \\ $$$$\begin{cases}{{x}={x}'\mathrm{cos}\:\alpha\:−{y}'\mathrm{sin}\:\alpha}\\{{y}={x}'\mathrm{sin}\:\alpha\:+{y}'\mathrm{cos}\:\alpha}\end{cases} \\ $$$$\mathrm{we}\:\mathrm{now}\:\mathrm{have}\:\left[\mathrm{using}\:{x},\:{y}\:\mathrm{again}\:\mathrm{instead}\:\mathrm{of}\:{x}',\:{y}'\right] \\ $$$${Ax}^{\mathrm{2}} +{Cy}^{\mathrm{2}} +{Dx}+{Ey}+{F}=\mathrm{0} \\ $$$$\mathrm{now}\:\mathrm{complete}\:\mathrm{the}\:\mathrm{squares} \\ $$$${A}\left({x}+\frac{{D}}{\mathrm{2}{A}}\right)^{\mathrm{2}} +{C}\left({y}+\frac{{E}}{\mathrm{2}{C}}\right)^{\mathrm{2}} +\left({F}−\frac{{D}^{\mathrm{2}} }{\mathrm{4}{A}^{\mathrm{2}} }−\frac{{E}^{\mathrm{2}} }{\mathrm{4}{C}^{\mathrm{2}} }\right)=\mathrm{0} \\ $$$$\begin{cases}{{x}={x}'−\frac{{D}}{\mathrm{2}{A}}}\\{{y}={y}'−\frac{{E}}{\mathrm{2}{C}}}\end{cases} \\ $$$$\mathrm{we}\:\mathrm{now}\:\mathrm{have}\:\left[\mathrm{using}\:{x},\:{y}\:\mathrm{again}\:\mathrm{instead}\:\mathrm{of}\:{x}',\:{y}'\right] \\ $$$$\mathrm{one}\:\mathrm{of}\:\mathrm{these} \\ $$$$\begin{cases}{\mathcal{A}{x}^{\mathrm{2}} +\mathcal{B}{y}^{\mathrm{2}} +\mathcal{C}=\mathrm{0}}\\{\mathcal{A}{x}+\mathcal{B}{y}^{\mathrm{2}} +\mathcal{C}=\mathrm{0}}\\{\mathcal{A}{x}^{\mathrm{2}} +\mathcal{B}{y}+\mathcal{C}=\mathrm{0}}\end{cases} \\ $$$$ \\ $$$${a},\:{c},\:{A},\:{C},\:\mathcal{A},\:\mathcal{B},\:\mathcal{C}\:\neq\mathrm{0} \\ $$$$\mathrm{in}\:\mathrm{all}\:\mathrm{other}\:\mathrm{cases}\:\mathrm{use}\:\mathrm{your}\:\mathrm{brain}\:\mathrm{to}\:\mathrm{interprete} \\ $$$$\mathrm{which}\:\mathrm{kind}\:\mathrm{of}\:\mathrm{curve}\:\left(\mathrm{if}\:\mathrm{any}\right)\:\mathrm{we}\:\mathrm{have} \\ $$

Question Number 78233 Answers: 0 Comments: 8

given 5x+22y=18 find for x,y integer

$${given}\:\mathrm{5}{x}+\mathrm{22}{y}=\mathrm{18} \\ $$$${find}\:{for}\:{x},{y}\:{integer} \\ $$

Question Number 78216 Answers: 1 Comments: 5

Question Number 78207 Answers: 1 Comments: 6

what minimum value of f(x)= 9tan^2 (x)+4cot^2 (x)

$${what}\:{minimum}\: \\ $$$${value}\:{of}\:{f}\left({x}\right)=\:\mathrm{9tan}\:^{\mathrm{2}} \left({x}\right)+\mathrm{4cot}\:^{\mathrm{2}} \left({x}\right) \\ $$

Question Number 78198 Answers: 1 Comments: 4

Question Number 78177 Answers: 1 Comments: 10

what equation of ellips with F_1 (1,2) F_2 (3,4) and a = (√3)

$${what}\:{equation}\:{of}\:{ellips} \\ $$$${with}\:{F}_{\mathrm{1}} \left(\mathrm{1},\mathrm{2}\right)\:{F}_{\mathrm{2}} \left(\mathrm{3},\mathrm{4}\right)\:{and}\:{a}\:=\:\sqrt{\mathrm{3}} \\ $$

Question Number 78168 Answers: 0 Comments: 6

Solve for x, y, z if: x^3 + y^3 + z^3 = 42

$$\mathrm{Solve}\:\mathrm{for}\:\:\mathrm{x},\:\mathrm{y},\:\mathrm{z}\:\:\mathrm{if}:\:\:\mathrm{x}^{\mathrm{3}} \:+\:\mathrm{y}^{\mathrm{3}} \:+\:\mathrm{z}^{\mathrm{3}} \:\:=\:\:\mathrm{42} \\ $$

Question Number 78163 Answers: 0 Comments: 2

∫ (√(1 + 3 sin(θ) + sin^2 (θ))) dθ

$$\int\:\sqrt{\mathrm{1}\:+\:\mathrm{3}\:\mathrm{sin}\left(\theta\right)\:+\:\mathrm{sin}^{\mathrm{2}} \left(\theta\right)}\:\:\mathrm{d}\theta \\ $$

Question Number 78162 Answers: 0 Comments: 4

Find the sum of nth term Σ_(k = 1) ^n (1/k^2 )

$$\mathrm{Find}\:\mathrm{the}\:\mathrm{sum}\:\mathrm{of}\:\mathrm{nth}\:\mathrm{term} \\ $$$$\:\:\:\underset{\mathrm{k}\:=\:\mathrm{1}} {\overset{\mathrm{n}} {\sum}}\:\frac{\mathrm{1}}{\mathrm{k}^{\mathrm{2}} } \\ $$

Question Number 78161 Answers: 1 Comments: 0

Question Number 78160 Answers: 0 Comments: 1

find the term independent of x in [ ((x−1)/x)]^9

$$\mathrm{find}\:\mathrm{the}\:\mathrm{term}\:\mathrm{independent}\:\mathrm{of}\:\mathrm{x}\:\mathrm{in} \\ $$$$\:\:\left[\:\:\frac{\mathrm{x}−\mathrm{1}}{\mathrm{x}}\right]^{\mathrm{9}} \\ $$

Question Number 78156 Answers: 0 Comments: 1

Given that u_(n + 1) = (a_n /2) + 5 evalatuate lim_(x→∞) a_n deduce if a_(n ) is convergent or divergent.

$$\mathrm{Given}\:\mathrm{that}\:{u}_{{n}\:+\:\mathrm{1}} =\:\frac{{a}_{{n}} }{\mathrm{2}}\:+\:\mathrm{5}\:\:\: \\ $$$${evalatuate}\:\:\underset{{x}\rightarrow\infty} {\mathrm{lim}}\:{a}_{{n}} \\ $$$${deduce}\:{if}\:{a}_{{n}\:} \:{is}\:{convergent}\:{or}\:{divergent}. \\ $$

Question Number 78153 Answers: 1 Comments: 1

Question Number 78147 Answers: 0 Comments: 0

Question Number 78136 Answers: 1 Comments: 5

give the equation of tangente at p(x_0 ,f(x_0 )) 1)f(x)=e^(−x^2 ) ln(1−2x) x_0 =−1 2)f(x)=(x^2 −3)arctan(x^2 ) x_0 =1 3) f(x) =((e^(−x) sin(πx))/(x^2 +3)) x_0 =(1/2) 4) f(x)=e^(−x) (√(e^x^2 −1)) and x_0 = 2

$${give}\:{the}\:{equation}\:{of}\:{tangente} \\ $$$${at}\:\:{p}\left({x}_{\mathrm{0}} ,{f}\left({x}_{\mathrm{0}} \right)\right) \\ $$$$\left.\mathrm{1}\right){f}\left({x}\right)={e}^{−{x}^{\mathrm{2}} } {ln}\left(\mathrm{1}−\mathrm{2}{x}\right)\:\:\:{x}_{\mathrm{0}} =−\mathrm{1} \\ $$$$\left.\mathrm{2}\right){f}\left({x}\right)=\left({x}^{\mathrm{2}} −\mathrm{3}\right){arctan}\left({x}^{\mathrm{2}} \right) \\ $$$${x}_{\mathrm{0}} \:\:=\mathrm{1} \\ $$$$\left.\mathrm{3}\right)\:{f}\left({x}\right)\:=\frac{{e}^{−{x}} {sin}\left(\pi{x}\right)}{{x}^{\mathrm{2}} \:+\mathrm{3}}\:\:\:{x}_{\mathrm{0}} =\frac{\mathrm{1}}{\mathrm{2}} \\ $$$$\left.\mathrm{4}\right)\:{f}\left({x}\right)={e}^{−{x}} \sqrt{{e}^{{x}^{\mathrm{2}} } −\mathrm{1}} \\ $$$${and}\:{x}_{\mathrm{0}} =\:\mathrm{2} \\ $$$$ \\ $$

Question Number 78135 Answers: 0 Comments: 0

explicit f(x)=∫_0 ^∞ ((arctan(xt))/(t^2 +x^2 ))dt with x>0

$${explicit}\:{f}\left({x}\right)=\int_{\mathrm{0}} ^{\infty} \:\:\frac{{arctan}\left({xt}\right)}{{t}^{\mathrm{2}} +{x}^{\mathrm{2}} }{dt} \\ $$$${with}\:{x}>\mathrm{0} \\ $$

Question Number 78134 Answers: 0 Comments: 1

find the value of ∫_(−∞) ^(+∞) ((arctan(3x^2 ))/(x^2 +4))dx

$${find}\:{the}\:{value}\:{of}\:\:\int_{−\infty} ^{+\infty} \:\frac{{arctan}\left(\mathrm{3}{x}^{\mathrm{2}} \right)}{{x}^{\mathrm{2}} +\mathrm{4}}{dx} \\ $$

Question Number 78122 Answers: 1 Comments: 3

lim_(x→1^− ) lnx∙ln(1−x)=?

$$\underset{{x}\rightarrow\mathrm{1}^{−} } {\mathrm{lim}}{lnx}\centerdot{ln}\left(\mathrm{1}−{x}\right)=? \\ $$

Question Number 78119 Answers: 1 Comments: 6

find the center point ellips 5x^2 +8y^2 −24x−24y+4xy=0