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Question Number 66107 Answers: 0 Comments: 3

Given that S_n = ((a(1 −r^n ))/(1−r)) , r ≠ 1, show that ((S_(3n) −S_(2n) )/(S_n )) = r^(2n) hence given that r =(1/2) find Σ_(n=0) ^∞ (((S_(3n) −S_(2n) )/S_n ))

$${Given}\:{that}\:{S}_{{n}} \:=\:\frac{{a}\left(\mathrm{1}\:−{r}^{{n}} \right)}{\mathrm{1}−{r}}\:,\:{r}\:\neq\:\mathrm{1},\:{show}\:{that}\:\frac{{S}_{\mathrm{3}{n}} \:−{S}_{\mathrm{2}{n}} }{{S}_{{n}} \:}\:=\:{r}^{\mathrm{2}{n}} \\ $$$${hence}\:{given}\:{that}\:{r}\:=\frac{\mathrm{1}}{\mathrm{2}}\:{find}\:\underset{{n}=\mathrm{0}} {\overset{\infty} {\sum}}\left(\frac{{S}_{\mathrm{3}{n}} \:−{S}_{\mathrm{2}{n}} }{{S}_{{n}} }\right) \\ $$

Question Number 66106 Answers: 1 Comments: 0

In an electrolysis experiment, the ammeter records a steady current of 1A. The mass of copper deposited in 30minutes is 0.66g. Calculate the error in the ammeter reading. the electrical equivalent of cu is 3.3×10^(−4) g/C.

$$\mathrm{In}\:\mathrm{an}\:\mathrm{electrolysis}\:\mathrm{experiment},\:\mathrm{the}\:\mathrm{ammeter} \\ $$$$\mathrm{records}\:\mathrm{a}\:\mathrm{steady}\:\mathrm{current}\:\mathrm{of}\:\mathrm{1A}.\:\mathrm{The}\:\mathrm{mass} \\ $$$$\mathrm{of}\:\mathrm{copper}\:\mathrm{deposited}\:\mathrm{in}\:\mathrm{30minutes}\:\mathrm{is}\:\mathrm{0}.\mathrm{66g}. \\ $$$$\mathrm{Calculate}\:\mathrm{the}\:\mathrm{error}\:\mathrm{in}\:\mathrm{the}\:\mathrm{ammeter}\:\mathrm{reading}. \\ $$$$\:\:\:\mathrm{the}\:\mathrm{electrical}\:\mathrm{equivalent}\:\mathrm{of}\:\mathrm{cu}\:\mathrm{is}\:\mathrm{3}.\mathrm{3}×\mathrm{10}^{−\mathrm{4}} \mathrm{g}/\mathrm{C}. \\ $$

Question Number 66105 Answers: 0 Comments: 3

Find ∫_(−1) ^1 ((9 +4x^2 )/(9−4x^2 )) dx

$${Find}\:\:\int_{−\mathrm{1}} ^{\mathrm{1}} \frac{\mathrm{9}\:+\mathrm{4}{x}^{\mathrm{2}} }{\mathrm{9}−\mathrm{4}{x}^{\mathrm{2}} }\:{dx} \\ $$

Question Number 66104 Answers: 0 Comments: 1

f(x)= 2x^3 −x−4 show that the equation f(x) =0 has root between 1 and 2 show that the equation f(x) =0 can be written as x = (√(((2/x) +(1/2)))) use the iteration x_(n+1 ) = (√(((2/x_n ) +(1/2)) ,)) with x_0 = 1.385 to find to 3 decimal places the value of x_1 .

$${f}\left({x}\right)=\:\mathrm{2}{x}^{\mathrm{3}} −{x}−\mathrm{4} \\ $$$${show}\:{that}\:{the}\:{equation}\:{f}\left({x}\right)\:=\mathrm{0}\:{has}\:{root}\:{between}\:\mathrm{1}\:{and}\:\mathrm{2} \\ $$$${show}\:{that}\:{the}\:{equation}\:{f}\left({x}\right)\:=\mathrm{0}\:{can}\:{be}\:{written}\:{as}\: \\ $$$$\:\:{x}\:=\:\sqrt{\left(\frac{\mathrm{2}}{{x}}\:+\frac{\mathrm{1}}{\mathrm{2}}\right)} \\ $$$${use}\:{the}\:{iteration} \\ $$$$\:{x}_{{n}+\mathrm{1}\:} \:=\:\sqrt{\left(\frac{\mathrm{2}}{{x}_{{n}} }\:+\frac{\mathrm{1}}{\mathrm{2}}\right)\:,} \\ $$$${with}\:{x}_{\mathrm{0}} \:=\:\mathrm{1}.\mathrm{385}\:{to}\:{find}\:{to}\:\mathrm{3}\:{decimal}\:{places}\:{the}\:{value}\:{of}\:{x}_{\mathrm{1}} . \\ $$$$ \\ $$

Question Number 66103 Answers: 0 Comments: 2

A binary relation R is defined on N,the set of natural numbers by _x R_y ⇔ ∃ n ∈ Z : x = 2^n y, x,y ∈ N show that R is an equivalence relation

$${A}\:{binary}\:{relation}\:{R}\:{is}\:{defined}\:{on}\:\mathbb{N},{the}\:{set}\:{of}\:{natural}\:{numbers}\:{by}\: \\ $$$$\:_{{x}} {R}_{{y}} \:\Leftrightarrow\:\exists\:{n}\:\in\:\mathbb{Z}\::\:{x}\:=\:\mathrm{2}^{{n}} {y},\:\:{x},{y}\:\in\:\mathbb{N} \\ $$$${show}\:{that}\:{R}\:{is}\:{an}\:{equivalence}\:{relation} \\ $$

Question Number 66102 Answers: 0 Comments: 3

prove by mathematical induction that 4^n +3^n +2 is a multiple of 3 for all positive integral values of n.

$${prove}\:{by}\:{mathematical}\:{induction}\:{that}\:\:\mathrm{4}^{{n}} +\mathrm{3}^{{n}} +\mathrm{2}\:{is}\:{a}\:{multiple}\:{of}\:\mathrm{3}\:{for}\:{all}\: \\ $$$${positive}\:{integral}\:{values}\:{of}\:{n}. \\ $$

Question Number 66101 Answers: 1 Comments: 1

find (dy/dx) when y = x^2 ln(3x) Given that xsinx − y^2 =0 show that y^2 = 2cosx −2((dy/dx))^2 −2y(d^2 y/dx^2 )

$${find}\:\frac{{dy}}{{dx}}\:\:{when}\:{y}\:=\:{x}^{\mathrm{2}} {ln}\left(\mathrm{3}{x}\right) \\ $$$${Given}\:{that}\:{xsinx}\:−\:{y}^{\mathrm{2}} =\mathrm{0}\:{show}\:{that}\:\:{y}^{\mathrm{2}} \:=\:\mathrm{2}{cosx}\:−\mathrm{2}\left(\frac{{dy}}{{dx}}\right)^{\mathrm{2}} \:−\mathrm{2}{y}\frac{{d}^{\mathrm{2}} {y}}{{dx}^{\mathrm{2}} } \\ $$

Question Number 66099 Answers: 0 Comments: 1

differentiate y=10^(1−sin^2 3x)

$${differentiate}\:{y}=\mathrm{10}^{\mathrm{1}−{sin}^{\mathrm{2}} \mathrm{3}{x}} \\ $$

Question Number 66098 Answers: 0 Comments: 1

Question Number 66096 Answers: 0 Comments: 2

∫ ((√((1+x^2 )))/x^2 ) dx = ?

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

Question Number 66090 Answers: 1 Comments: 0

∫_( 0) ^(π/2) ((sin^2 x)/(sin x+cos x)) dx =

$$\:\underset{\:\mathrm{0}} {\overset{\pi/\mathrm{2}} {\int}}\:\frac{\mathrm{sin}^{\mathrm{2}} {x}}{\mathrm{sin}\:{x}+\mathrm{cos}\:{x}}\:{dx}\:= \\ $$

Question Number 66089 Answers: 0 Comments: 0

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Question Number 66085 Answers: 0 Comments: 0

calculate ∫_0 ^1 ((arctan((√(x^2 +2))))/((x^2 +1)(√(x^2 +2))))dx

$${calculate}\:\int_{\mathrm{0}} ^{\mathrm{1}} \:\frac{{arctan}\left(\sqrt{{x}^{\mathrm{2}} +\mathrm{2}}\right)}{\left({x}^{\mathrm{2}} \:+\mathrm{1}\right)\sqrt{{x}^{\mathrm{2}} \:+\mathrm{2}}}{dx} \\ $$

Question Number 66084 Answers: 1 Comments: 1

How to solve this limit? lim_(x→∞) (7x+(2/x))^x

$${How}\:{to}\:{solve}\:{this}\:{limit}? \\ $$$$\underset{{x}\rightarrow\infty} {{lim}}\left(\mathrm{7}{x}+\frac{\mathrm{2}}{{x}}\right)^{{x}} \\ $$

Question Number 66082 Answers: 0 Comments: 3

Question Number 66071 Answers: 1 Comments: 0

Evaluate the integral as a limit of sums: 1.∫_1 ^3 (e^(2−3x) +x^2 +1)dx

$$\mathrm{Evaluate}\:\mathrm{the}\:\mathrm{integral}\:\mathrm{as}\:\mathrm{a}\:\mathrm{limit}\:\mathrm{of}\:\mathrm{sums}: \\ $$$$\mathrm{1}.\int_{\mathrm{1}} ^{\mathrm{3}} \left({e}^{\mathrm{2}−\mathrm{3}{x}} +{x}^{\mathrm{2}} +\mathrm{1}\right){dx} \\ $$

Question Number 66066 Answers: 0 Comments: 0

Integrate ∫(dx/(ax^2 +bx+c)) y = ax^2 +bx+c y′=(dy/dx)=2ax+b d = (√(b^2 −4ac)) d′ = (√(−d)) Case 1. d^2 < 0 I=(2/d′)tan^(−1) ((y′)/d′)+C [tan^(−1) α=arctan α] Case 2. d^2 =0 I=((−2)/(y′))+C Case 3. d^2 > 0 I=(1/d)ln∣((y′−d)/(y′+d))∣+C [ln a = log_e a]

$${Integrate}\:\int\frac{{dx}}{{ax}^{\mathrm{2}} +{bx}+{c}} \\ $$$${y}\:=\:{ax}^{\mathrm{2}} +{bx}+{c} \\ $$$${y}'=\frac{{dy}}{{dx}}=\mathrm{2}{ax}+{b} \\ $$$${d}\:=\:\sqrt{{b}^{\mathrm{2}} −\mathrm{4}{ac}} \\ $$$${d}'\:=\:\sqrt{−{d}} \\ $$$${Case}\:\mathrm{1}.\:{d}^{\mathrm{2}} \:<\:\mathrm{0} \\ $$$${I}=\frac{\mathrm{2}}{{d}'}{tan}^{−\mathrm{1}} \frac{{y}'}{{d}'}+{C}\:\:\:\left[{tan}^{−\mathrm{1}} \alpha={arctan}\:\alpha\right] \\ $$$${Case}\:\mathrm{2}.\:{d}^{\mathrm{2}} =\mathrm{0} \\ $$$${I}=\frac{−\mathrm{2}}{{y}'}+{C} \\ $$$${Case}\:\mathrm{3}.\:{d}^{\mathrm{2}} \:>\:\mathrm{0} \\ $$$${I}=\frac{\mathrm{1}}{{d}}\mathrm{ln}\mid\frac{{y}'−{d}}{{y}'+{d}}\mid+{C}\:\:\:\left[\mathrm{ln}\:{a}\:=\:\mathrm{log}_{{e}} \:{a}\right] \\ $$

Question Number 66065 Answers: 0 Comments: 2

find the value of U_n =∫_(−∞) ^(+∞) e^(−nx^2 ) sin(x^2 −2x)dx find nature of the serie Σ U_n and Σe^(−n^2 ) U_n

$${find}\:{the}\:{value}\:{of}\:{U}_{{n}} =\int_{−\infty} ^{+\infty} {e}^{−{nx}^{\mathrm{2}} } {sin}\left({x}^{\mathrm{2}} −\mathrm{2}{x}\right){dx} \\ $$$${find}\:{nature}\:{of}\:{the}\:{serie}\:\Sigma\:{U}_{{n}} \:{and}\:\Sigma{e}^{−{n}^{\mathrm{2}} } {U}_{{n}} \\ $$

Question Number 66064 Answers: 0 Comments: 1

find the value of ∫_(−∞) ^(+∞) cos(x^2 −x+1)dx

$${find}\:{the}\:{value}\:{of}\:\int_{−\infty} ^{+\infty} \:{cos}\left({x}^{\mathrm{2}} −{x}+\mathrm{1}\right){dx} \\ $$

Question Number 66063 Answers: 0 Comments: 0

let x^2 −x +lnx =0 by using newton method find a approximate value of the roots of this equation.

$${let}\:\:\:{x}^{\mathrm{2}} −{x}\:+{lnx}\:=\mathrm{0}\:\:\:\:\:{by}\:{using}\:{newton}\:{method}\:{find} \\ $$$${a}\:{approximate}\:{value}\:{of}\:{the}\:{roots}\:{of}\:{this}\:{equation}. \\ $$$$ \\ $$

Question Number 66062 Answers: 0 Comments: 3

let f(x) =∫_0 ^1 (dt/(ch(t)+xsh(t))) 1) find a explicit form of f(x) 2) determine g(x) =∫_0 ^1 (dt/((ch(t)+xsh(t))^2 )) 3) calculate ∫_0 ^1 (dt/(ch(t)+3sh(t))) and ∫_0 ^1 (dt/({ch(t)+3sh(t)}^2 ))

$${let}\:{f}\left({x}\right)\:=\int_{\mathrm{0}} ^{\mathrm{1}} \:\:\:\frac{{dt}}{{ch}\left({t}\right)+{xsh}\left({t}\right)} \\ $$$$\left.\mathrm{1}\right)\:{find}\:{a}\:{explicit}\:{form}\:{of}\:{f}\left({x}\right) \\ $$$$\left.\mathrm{2}\right)\:{determine}\:{g}\left({x}\right)\:=\int_{\mathrm{0}} ^{\mathrm{1}} \:\:\frac{{dt}}{\left({ch}\left({t}\right)+{xsh}\left({t}\right)\right)^{\mathrm{2}} } \\ $$$$\left.\mathrm{3}\right)\:{calculate}\:\int_{\mathrm{0}} ^{\mathrm{1}} \:\:\:\frac{{dt}}{{ch}\left({t}\right)+\mathrm{3}{sh}\left({t}\right)}\:{and}\:\int_{\mathrm{0}} ^{\mathrm{1}} \:\:\frac{{dt}}{\left\{{ch}\left({t}\right)+\mathrm{3}{sh}\left({t}\right)\right\}^{\mathrm{2}} } \\ $$

Question Number 66061 Answers: 0 Comments: 0

let f(t) =∫_0 ^1 ((sinx)/(1+te^(−x^2 ) ))dx with ∣t∣<1 developp f at integr serie .

$${let}\:{f}\left({t}\right)\:=\int_{\mathrm{0}} ^{\mathrm{1}} \:\:\:\frac{{sinx}}{\mathrm{1}+{te}^{−{x}^{\mathrm{2}} } }{dx}\:\:\:\:{with}\:\mid{t}\mid<\mathrm{1} \\ $$$${developp}\:{f}\:{at}\:{integr}\:{serie}\:. \\ $$

Question Number 66060 Answers: 0 Comments: 3

let f(x) =∫_0 ^(π/4) (dt/(x+tant)) with x real 1) find aexplicit form of f(x) 2)find also g(x) =∫_0 ^(π/4) (dt/((x+tant)^2 )) 3)give f^((n)) (x)at form of integral 4)calculate ∫_0 ^(π/4) (dt/(2+tant)) and ∫_0 ^(π/4) (dt/((2+tant)^2 ))

$${let}\:{f}\left({x}\right)\:=\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{4}}} \:\:\frac{{dt}}{{x}+{tant}}\:\:{with}\:{x}\:{real} \\ $$$$\left.\mathrm{1}\right)\:{find}\:{aexplicit}\:{form}\:{of}\:{f}\left({x}\right) \\ $$$$\left.\mathrm{2}\right){find}\:{also}\:{g}\left({x}\right)\:=\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{4}}} \:\:\frac{{dt}}{\left({x}+{tant}\right)^{\mathrm{2}} } \\ $$$$\left.\mathrm{3}\right){give}\:{f}^{\left({n}\right)} \left({x}\right){at}\:{form}\:{of}\:{integral} \\ $$$$\left.\mathrm{4}\right){calculate}\:\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{4}}} \:\:\frac{{dt}}{\mathrm{2}+{tant}}\:\:{and}\:\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{4}}} \:\:\frac{{dt}}{\left(\mathrm{2}+{tant}\right)^{\mathrm{2}} } \\ $$

Question Number 66058 Answers: 2 Comments: 5

If x + (1/x)=1 find out value:− ((x^(20) +x^(17) +x^(14) +x^(11) )/(x^(17) +x^(14) +x^(11) +x^8 )) = ?

$${If}\:\:\:\:{x}\:+\:\frac{\mathrm{1}}{{x}}=\mathrm{1} \\ $$$$ \\ $$$${find}\:{out}\:{value}:− \\ $$$$ \\ $$$$\:\:\:\:\frac{{x}^{\mathrm{20}} +{x}^{\mathrm{17}} +{x}^{\mathrm{14}} +{x}^{\mathrm{11}} }{{x}^{\mathrm{17}} +{x}^{\mathrm{14}} +{x}^{\mathrm{11}} +{x}^{\mathrm{8}} }\:\:\:\:=\:\:\:\:? \\ $$

Question Number 66055 Answers: 1 Comments: 1

Question Number 66048 Answers: 0 Comments: 1

∫(x/(√(ln(1/x)))) dx

$$\int\frac{{x}}{\sqrt{{ln}\left(\mathrm{1}/{x}\right)}}\:{dx} \\ $$

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