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Question Number 33157 Answers: 0 Comments: 1

find k if kloga= loga + log a^2 + log a^(3 ) + log a^4

$${find}\:{k}\:{if} \\ $$$${kloga}=\:{loga}\:+\:{log}\:{a}^{\mathrm{2}} +\:{log}\:{a}^{\mathrm{3}\:} +\:{log}\:{a}^{\mathrm{4}} \\ $$

Question Number 33155 Answers: 0 Comments: 4

Evaluate ∫_(−∞) ^∞ 3x^2 (x^3 + 1)^2 e^(−x^6 − 2x^3 ) dx

$$\mathrm{Evaluate} \\ $$$$\int_{−\infty} ^{\infty} \:\mathrm{3}{x}^{\mathrm{2}} \left({x}^{\mathrm{3}} \:+\:\mathrm{1}\right)^{\mathrm{2}} \:{e}^{−{x}^{\mathrm{6}} \:−\:\mathrm{2}{x}^{\mathrm{3}} } \:{dx} \\ $$

Question Number 33154 Answers: 0 Comments: 1

it is given that Σ_(r=1) ^n U_n = ((1+3^(2n+2) −2×5^(n+1) )/8) where U_n is the n^(th) term of a sequence find the simplified expression for U_n

$${it}\:{is}\:{given}\:{that}\:\underset{{r}=\mathrm{1}} {\overset{{n}} {\sum}}\:{U}_{{n}} =\:\frac{\mathrm{1}+\mathrm{3}^{\mathrm{2}{n}+\mathrm{2}} −\mathrm{2}×\mathrm{5}^{{n}+\mathrm{1}} }{\mathrm{8}} \\ $$$${where}\:{U}_{{n}} \:{is}\:{the}\:{n}^{{th}} \:{term}\:{of}\:{a}\:{sequence} \\ $$$${find}\:{the}\:{simplified}\:{expression}\:{for}\:{U}_{{n}} \\ $$

Question Number 33151 Answers: 0 Comments: 0

Question Number 33147 Answers: 2 Comments: 1

If x^m occurs in the expansion of (x + (1/x^2 ))^(2n) , the coefficient of x^m is

$$\mathrm{If}\:\:\:{x}^{{m}} \:\:\mathrm{occurs}\:\mathrm{in}\:\mathrm{the}\:\mathrm{expansion}\:\mathrm{of} \\ $$$$\left({x}\:+\:\frac{\mathrm{1}}{{x}^{\mathrm{2}} }\right)^{\mathrm{2}{n}} ,\:\mathrm{the}\:\mathrm{coefficient}\:\mathrm{of}\:{x}^{{m}} \:\mathrm{is} \\ $$

Question Number 33146 Answers: 0 Comments: 0

If the expression exp{(1+∣cos x∣+cos^2 x+∣cos^3 x∣+cos^4 x +....∞)log_e 4} satisfies the equation y^2 −20y+64=0 for 0< x <π, then the set of values of x is

$$\mathrm{If}\:\mathrm{the}\:\mathrm{expression}\: \\ $$$${exp}\left\{\left(\mathrm{1}+\mid\mathrm{cos}\:{x}\mid+\mathrm{cos}^{\mathrm{2}} {x}+\mid\mathrm{cos}^{\mathrm{3}} {x}\mid+\mathrm{cos}^{\mathrm{4}} {x}\right.\right. \\ $$$$\left.\:\left.\:\:\:\:\:\:\:\:\:\:\:+....\infty\right)\mathrm{log}_{{e}} \mathrm{4}\right\}\:\mathrm{satisfies}\:\mathrm{the}\:\mathrm{equation} \\ $$$${y}^{\mathrm{2}} −\mathrm{20}{y}+\mathrm{64}=\mathrm{0}\:\mathrm{for}\:\mathrm{0}<\:{x}\:<\pi,\:\mathrm{then}\:\mathrm{the}\:\mathrm{set} \\ $$$$\mathrm{of}\:\mathrm{values}\:\mathrm{of}\:{x}\:\mathrm{is} \\ $$

Question Number 33145 Answers: 1 Comments: 1

∫_( 0) ^3 (dx/((√(x+1)) + (√(5x+1)))) =

$$\underset{\:\mathrm{0}} {\overset{\mathrm{3}} {\int}}\:\frac{{dx}}{\sqrt{{x}+\mathrm{1}}\:+\:\sqrt{\mathrm{5}{x}+\mathrm{1}}}\:= \\ $$

Question Number 33139 Answers: 1 Comments: 0

α and β are roots of ax^2 +bx +c=0 show that α+β= ((−b)/a) and αβ= (c/a) hence form an equation whose sum of roots and product of roots are respectively −(1/2) and 2.

$$\alpha\:{and}\:\beta\:{are}\:{roots}\:{of}\: \\ $$$$\:{ax}^{\mathrm{2}} +{bx}\:+{c}=\mathrm{0} \\ $$$${show}\:{that}\:\alpha+\beta=\:\frac{−{b}}{{a}}\:{and}\:\alpha\beta=\:\frac{{c}}{{a}} \\ $$$${hence}\:{form}\:{an}\:{equation}\:{whose} \\ $$$${sum}\:{of}\:{roots}\:{and}\:{product}\:{of}\:{roots} \\ $$$${are}\:{respectively}\: \\ $$$$\:\:\:−\frac{\mathrm{1}}{\mathrm{2}}\:{and}\:\mathrm{2}. \\ $$

Question Number 33136 Answers: 1 Comments: 0

given that y= 3x^4 find the percentage increase in y when x increases at (5/2)%.

$${given}\:{that}\:{y}=\:\mathrm{3}{x}^{\mathrm{4}} \:{find}\:{the}\:{percentage} \\ $$$${increase}\:{in}\:{y}\:{when}\:{x}\:{increases}\:{at}\:\frac{\mathrm{5}}{\mathrm{2}}\%. \\ $$$$ \\ $$

Question Number 33131 Answers: 0 Comments: 1

1)find Σ_(n=1) ^∞ (e^(inx) /(n(n+1))) 2) find the value of Σ_(n≥1) ((sin(nx))/(n(n+1))) and Σ_(n≥1) ((cos(nx))/(n(n+1))) .

$$\left.\mathrm{1}\right){find}\:\:\sum_{{n}=\mathrm{1}} ^{\infty} \:\:\frac{{e}^{{inx}} }{{n}\left({n}+\mathrm{1}\right)} \\ $$$$\left.\mathrm{2}\right)\:{find}\:{the}\:{value}\:{of}\:\:\:\sum_{{n}\geqslant\mathrm{1}} \:\frac{{sin}\left({nx}\right)}{{n}\left({n}+\mathrm{1}\right)} \\ $$$${and}\:\sum_{{n}\geqslant\mathrm{1}} \:\:\frac{{cos}\left({nx}\right)}{{n}\left({n}+\mathrm{1}\right)}\:. \\ $$

Question Number 33130 Answers: 0 Comments: 0

find the value of ∫_0 ^∞ ((1+x cosθ)/(x^2 +2x cosθ +1)) dx .

$${find}\:\:{the}\:{value}\:{of}\:\:\int_{\mathrm{0}} ^{\infty} \:\:\frac{\mathrm{1}+{x}\:{cos}\theta}{{x}^{\mathrm{2}} \:+\mathrm{2}{x}\:{cos}\theta\:+\mathrm{1}}\:{dx}\:. \\ $$

Question Number 33129 Answers: 0 Comments: 2

1)find the value of u_n =∫_(−∞) ^(+∞) ((cos(nx))/(4 +x^2 )) dx 2) find the nature of Σ u_n .

$$\left.\mathrm{1}\right){find}\:{the}\:{value}\:{of}\:\:\:{u}_{{n}} =\int_{−\infty} ^{+\infty} \:\:\:\:\frac{{cos}\left({nx}\right)}{\mathrm{4}\:+{x}^{\mathrm{2}} }\:{dx} \\ $$$$\left.\mathrm{2}\right)\:{find}\:{the}\:{nature}\:{of}\:\Sigma\:{u}_{{n}} \:. \\ $$

Question Number 33128 Answers: 0 Comments: 2

find the value of ∫_0 ^∞ (dx/((1+x^2 )( 1+x^4 ))) .

$${find}\:{the}\:{value}\:{of}\:\:\int_{\mathrm{0}} ^{\infty} \:\:\:\:\:\:\:\frac{{dx}}{\left(\mathrm{1}+{x}^{\mathrm{2}} \right)\left(\:\mathrm{1}+{x}^{\mathrm{4}} \right)}\:. \\ $$

Question Number 33127 Answers: 0 Comments: 1

find Σ_(n=0) ^∞ ((sin(na))/((sina)^n )) (x^n /(n!)) with 0<a<π .

$$\:{find}\:\:\sum_{{n}=\mathrm{0}} ^{\infty} \:\:\frac{{sin}\left({na}\right)}{\left({sina}\right)^{{n}} }\:\frac{{x}^{{n}} }{{n}!}\:\:{with}\:\mathrm{0}<{a}<\pi\:. \\ $$

Question Number 33126 Answers: 0 Comments: 1

let give f(x)= (1/(2x^2 −3x+1)) 1) find f^((n)) (x) 2) find f^((n)) (0) 3) if f(x)=Σ a_n x^n calculate the sequence a_n

$${let}\:{give}\:{f}\left({x}\right)=\:\frac{\mathrm{1}}{\mathrm{2}{x}^{\mathrm{2}} \:−\mathrm{3}{x}+\mathrm{1}} \\ $$$$\left.\mathrm{1}\right)\:{find}\:\:{f}^{\left({n}\right)} \left({x}\right) \\ $$$$\left.\mathrm{2}\right)\:{find}\:{f}^{\left({n}\right)} \left(\mathrm{0}\right) \\ $$$$\left.\mathrm{3}\right)\:{if}\:\:\:\:{f}\left({x}\right)=\Sigma\:{a}_{{n}} \:{x}^{{n}} \:\:{calculate}\:{the}\:{sequence}\:{a}_{{n}} \\ $$

Question Number 33122 Answers: 0 Comments: 0

let considere f and u differenciable function prove that (d/dt)( ∫_a ^(u(t)) f(t,x)dx)=∫_a ^(u(t)) (∂f/∂t)(t,x)dx +f(t,u(t))u^′ (t)

$${let}\:{considere}\:{f}\:{and}\:{u}\:{differenciable}\:{function}\:{prove} \\ $$$${that}\:\frac{{d}}{{dt}}\left(\:\int_{{a}} ^{{u}\left({t}\right)} {f}\left({t},{x}\right){dx}\right)=\int_{{a}} ^{{u}\left({t}\right)} \:\frac{\partial{f}}{\partial{t}}\left({t},{x}\right){dx}\:+{f}\left({t},{u}\left({t}\right)\right){u}^{'} \left({t}\right) \\ $$

Question Number 33120 Answers: 1 Comments: 0

let give α>0 find the value of ∫_0 ^1 (dx/(√((1−x)(1+αx)))) .

$${let}\:{give}\:\alpha>\mathrm{0}\:{find}\:{the}\:{value}\:{of}\:\:\int_{\mathrm{0}} ^{\mathrm{1}} \:\:\:\:\frac{{dx}}{\sqrt{\left(\mathrm{1}−{x}\right)\left(\mathrm{1}+\alpha{x}\right)}}\:. \\ $$

Question Number 33119 Answers: 0 Comments: 1

find ∫_0 ^∞ (t^n /(e^t −1)) dt by using ξ(x) for n integr ξ(x)=Σ_(n=1) ^∞ (1/n^x ) with x>1 .

$${find}\:\:\int_{\mathrm{0}} ^{\infty} \:\:\:\frac{{t}^{{n}} }{{e}^{{t}} \:−\mathrm{1}}\:{dt}\:{by}\:{using}\:\xi\left({x}\right)\:{for}\:{n}\:{integr} \\ $$$$\xi\left({x}\right)=\sum_{{n}=\mathrm{1}} ^{\infty} \:\frac{\mathrm{1}}{{n}^{{x}} }\:\:\:{with}\:{x}>\mathrm{1}\:. \\ $$

Question Number 33116 Answers: 1 Comments: 0

solve at [0,π] cosα +cos(2α) +cos(3α)=0

$${solve}\:{at}\:\left[\mathrm{0},\pi\right]\:\:{cos}\alpha\:+{cos}\left(\mathrm{2}\alpha\right)\:+{cos}\left(\mathrm{3}\alpha\right)=\mathrm{0} \\ $$

Question Number 33134 Answers: 0 Comments: 0

describe geometrically the transformation with matrix 1) (((1 0)),((0 −1)) ) 2) (((1 0)),((2 1)) )

$${describe}\:{geometrically}\:{the}\:{transformation} \\ $$$${with}\:{matrix} \\ $$$$\left.\mathrm{1}\left.\right)\:\begin{pmatrix}{\mathrm{1}\:\:\:\:\:\:\:\:\mathrm{0}}\\{\mathrm{0}\:\:\:\:\:\:\:−\mathrm{1}}\end{pmatrix}\:\:\:\mathrm{2}\right)\:\:\begin{pmatrix}{\mathrm{1}\:\:\:\:\:\:\:\:\:\mathrm{0}}\\{\mathrm{2}\:\:\:\:\:\:\:\:\:\mathrm{1}}\end{pmatrix} \\ $$

Question Number 33103 Answers: 1 Comments: 1

Question Number 33100 Answers: 1 Comments: 0

Question Number 33135 Answers: 0 Comments: 0

the triangle with vertices A(−1,−3) ,B(2,1),and C(−2,2), is transformed by matrix (((a b)),((c d)) ) into the triangle with vertices A(−2,−3)^ , B(4,1),C(−4^ ,2) find the values of a,b,c and d

$${the}\:{triangle}\:{with}\:{vertices}\:{A}\left(−\mathrm{1},−\mathrm{3}\right) \\ $$$$,{B}\left(\mathrm{2},\mathrm{1}\right),{and}\:{C}\left(−\mathrm{2},\mathrm{2}\right),\:{is}\:{transformed} \\ $$$${by}\:{matrix}\:\begin{pmatrix}{{a}\:\:\:\:\:\:{b}}\\{{c}\:\:\:\:\:\:\:{d}}\end{pmatrix}\:{into}\:{the}\:{triangle}\:{with}\:{vertices} \\ $$$${A}\left(−\mathrm{2},−\mathrm{3}\bar {\right)},\:{B}\left(\mathrm{4},\mathrm{1}\right),{C}\left(−\bar {\mathrm{4}},\mathrm{2}\right)\:{find}\:{the}\:{values}\:{of}\: \\ $$$${a},{b},{c}\:{and}\:{d} \\ $$$$ \\ $$

Question Number 33096 Answers: 1 Comments: 0

z and w ∈ C proof ∣∣z∣−∣w∣∣ ≤ ∣z−w∣ and ∣z∣−∣w∣≤ ∣z+w∣

$${z}\:{and}\:{w}\:\in\:\mathbb{C} \\ $$$${proof}\:\mid\mid{z}\mid−\mid{w}\mid\mid\:\leqslant\:\mid{z}−{w}\mid\:{and}\:\mid{z}\mid−\mid{w}\mid\leqslant\:\mid{z}+{w}\mid \\ $$

Question Number 33095 Answers: 1 Comments: 0

z=x+yi ∈ C z^− =x−y ∈ C proof ∣z∣^2 =∣z^2 ∣=zz^− , so z≠0 →(1/z)=(z^− /(∣z∣^2 ))

$${z}={x}+{yi}\:\in\:\mathbb{C} \\ $$$$\overset{−} {{z}}={x}−{y}\:\in\:\mathbb{C} \\ $$$${proof}\:\mid{z}\mid^{\mathrm{2}} =\mid{z}^{\mathrm{2}} \mid={z}\overset{−} {{z}},\:{so}\:{z}\neq\mathrm{0}\:\rightarrow\frac{\mathrm{1}}{{z}}=\frac{\overset{−} {{z}}}{\mid{z}\mid^{\mathrm{2}} } \\ $$

Question Number 33094 Answers: 0 Comments: 1

let f(x)= (1/(1+x+x^2 )) dvelopp f at integr serie.

$${let}\:{f}\left({x}\right)=\:\frac{\mathrm{1}}{\mathrm{1}+{x}+{x}^{\mathrm{2}} }\:\:{dvelopp}\:{f}\:{at}\:{integr}\:{serie}. \\ $$

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