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Question Number 68289 Answers: 1 Comments: 0

A circle is divided into two equal parts By An arc with center on the circle. Determine (a) The length of the arc (b)The ratio in which the arc divides the diameter meeting the center of the arc.

$$\mathrm{A}\:\mathrm{circle}\:\mathrm{is}\:\mathrm{divided}\:\mathrm{into}\:\mathrm{two}\:\mathrm{equal}\:\mathrm{parts} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\mathrm{By} \\ $$$$\:\mathrm{An}\:\mathrm{arc}\:\mathrm{with}\:\mathrm{center}\:\mathrm{on}\:\mathrm{the}\:\mathrm{circle}. \\ $$$$\mathcal{D}{etermine} \\ $$$$\:\:\left({a}\right)\:{The}\:{length}\:{of}\:{the}\:{arc} \\ $$$$\:\:\left({b}\right){The}\:{ratio}\:{in}\:{which}\:{the}\:{arc} \\ $$$$\:\:\:\:\:\:\:\:{divides}\:{the}\:{diameter}\: \\ $$$$\:\:\:\:\:\:\:\:{meeting}\:{the}\:{center}\:{of}\:{the}\:{arc}. \\ $$

Question Number 68272 Answers: 1 Comments: 0

Question Number 68271 Answers: 0 Comments: 0

Find J=∫_0 ^1 ((W(−ulnu))/(ulnu)) du when W is the lambert function

$$\:\:{Find}\:\:{J}=\int_{\mathrm{0}} ^{\mathrm{1}} \:\:\:\frac{{W}\left(−{ulnu}\right)}{{ulnu}}\:{du}\:\:\:\:{when}\:\:{W}\:{is}\:{the}\:{lambert}\:{function} \\ $$

Question Number 68270 Answers: 1 Comments: 3

Prove that if Li_2 (x)=Σ_(n=1) (x^n /n^2 ) then ∀ x Li_2 (x)+Li_2 (1−x) = (π^2 /6) −ln(x)ln(1−x) ∀ x∉[0:1] Li_2 (x)+Li_2 ((1/x)) = −(π^2 /6) −[ln(−x)]^2 Find A=Σ_(n=1) ^∞ (ϕ^n /n^2 ) and B=Σ_(n=1) ^∞ (2^n /n^2 )

$$\:{Prove}\:{that}\:\:{if}\:\:{Li}_{\mathrm{2}} \left({x}\right)=\underset{{n}=\mathrm{1}} {\sum}\:\frac{{x}^{{n}} }{{n}^{\mathrm{2}} }\:\:\:{then} \\ $$$$\forall\:{x}\:\:{Li}_{\mathrm{2}} \left({x}\right)+{Li}_{\mathrm{2}} \left(\mathrm{1}−{x}\right)\:=\:\frac{\pi^{\mathrm{2}} }{\mathrm{6}}\:−{ln}\left({x}\right){ln}\left(\mathrm{1}−{x}\right)\:\: \\ $$$$\forall\:{x}\notin\left[\mathrm{0}:\mathrm{1}\right]\:{Li}_{\mathrm{2}} \left({x}\right)+{Li}_{\mathrm{2}} \left(\frac{\mathrm{1}}{{x}}\right)\:=\:−\frac{\pi^{\mathrm{2}} }{\mathrm{6}}\:−\left[{ln}\left(−{x}\right)\right]^{\mathrm{2}} \:\: \\ $$$${Find}\:\:{A}=\underset{{n}=\mathrm{1}} {\overset{\infty} {\sum}}\:\frac{\varphi^{{n}} }{{n}^{\mathrm{2}} }\:\:{and}\:\:{B}=\underset{{n}=\mathrm{1}} {\overset{\infty} {\sum}}\:\frac{\mathrm{2}^{{n}} }{{n}^{\mathrm{2}} }\:\: \\ $$

Question Number 68260 Answers: 0 Comments: 1

find f(x) if f((1/x))+f(1−x)=x

$${find}\:{f}\left({x}\right)\:{if}\: \\ $$$${f}\left(\frac{\mathrm{1}}{{x}}\right)+{f}\left(\mathrm{1}−{x}\right)={x} \\ $$

Question Number 68278 Answers: 0 Comments: 2

Question Number 68257 Answers: 0 Comments: 0

Prove that (6/(673)) Σ_(n=1) ^∞ (1/(n^2 (((2n)),(n) ))) = (π^2 /(2019))

$$\:\:{Prove}\:{that}\:\:\frac{\mathrm{6}}{\mathrm{673}}\:\underset{{n}=\mathrm{1}} {\overset{\infty} {\sum}}\:\frac{\mathrm{1}}{{n}^{\mathrm{2}} \begin{pmatrix}{\mathrm{2}{n}}\\{{n}}\end{pmatrix}}\:=\:\frac{\pi^{\mathrm{2}} }{\mathrm{2019}}\: \\ $$

Question Number 68244 Answers: 0 Comments: 1

find nature of the serie Σ_(n=1) ^∞ arctan(n+(1/n))

$${find}\:{nature}\:{of}\:{the}\:{serie}\:\sum_{{n}=\mathrm{1}} ^{\infty} \:{arctan}\left({n}+\frac{\mathrm{1}}{{n}}\right) \\ $$

Question Number 68243 Answers: 0 Comments: 3

let f(x) =arctan(ax +1) with a real 1) calculate f^((n)) (x) and f^((n)) (0) 2) developp f at integr serie 3) calculate ∫_(−∞) ^(+∞) ((f(x))/(x^2 +4))dx

$${let}\:{f}\left({x}\right)\:={arctan}\left({ax}\:+\mathrm{1}\right)\:\:{with}\:{a}\:{real} \\ $$$$\left.\mathrm{1}\right)\:{calculate}\:{f}^{\left({n}\right)} \left({x}\right)\:{and}\:{f}^{\left({n}\right)} \left(\mathrm{0}\right) \\ $$$$\left.\mathrm{2}\right)\:{developp}\:{f}\:{at}\:{integr}\:{serie} \\ $$$$\left.\mathrm{3}\right)\:{calculate}\:\int_{−\infty} ^{+\infty} \:\frac{{f}\left({x}\right)}{{x}^{\mathrm{2}} \:+\mathrm{4}}{dx} \\ $$

Question Number 68242 Answers: 0 Comments: 0

find lim_(x→0) ((cos(πx^x )+1)/x^2 )

$${find}\:{lim}_{{x}\rightarrow\mathrm{0}} \:\:\:\frac{{cos}\left(\pi{x}^{{x}} \right)+\mathrm{1}}{{x}^{\mathrm{2}} } \\ $$

Question Number 68241 Answers: 0 Comments: 1

calculate ∫∫_w (x^2 −2y^2 )(√(x^2 +3y^2 ))dxdy with w ={(x,y)∈R^2 / 0≤x≤1 and 1≤y≤2}

$${calculate}\:\int\int_{{w}} \:\:\:\left({x}^{\mathrm{2}} −\mathrm{2}{y}^{\mathrm{2}} \right)\sqrt{{x}^{\mathrm{2}} +\mathrm{3}{y}^{\mathrm{2}} }{dxdy} \\ $$$${with}\:{w}\:=\left\{\left({x},{y}\right)\in{R}^{\mathrm{2}} /\:\mathrm{0}\leqslant{x}\leqslant\mathrm{1}\:\:{and}\:\mathrm{1}\leqslant{y}\leqslant\mathrm{2}\right\} \\ $$

Question Number 68240 Answers: 0 Comments: 1

calculate ∫_0 ^∞ ((arctan(3x)−arctan(2x))/x)dx

$${calculate}\:\:\int_{\mathrm{0}} ^{\infty} \:\:\frac{{arctan}\left(\mathrm{3}{x}\right)−{arctan}\left(\mathrm{2}{x}\right)}{{x}}{dx} \\ $$

Question Number 68239 Answers: 0 Comments: 2

let f(x) =e^(−2x) ln(1+x^2 ) developp f at integr serie.

$${let}\:{f}\left({x}\right)\:={e}^{−\mathrm{2}{x}} {ln}\left(\mathrm{1}+{x}^{\mathrm{2}} \right) \\ $$$${developp}\:{f}\:{at}\:{integr}\:{serie}. \\ $$

Question Number 68238 Answers: 0 Comments: 2

let f(x)=(x^2 −3x)arctan(2x+1) 1) determine f^((n)) (x) and f^((n)) (0) 2)developp f at integr serie 3) calculate ∫_0 ^1 f(x)dx

$${let}\:\:{f}\left({x}\right)=\left({x}^{\mathrm{2}} −\mathrm{3}{x}\right){arctan}\left(\mathrm{2}{x}+\mathrm{1}\right) \\ $$$$\left.\mathrm{1}\right)\:{determine}\:{f}^{\left({n}\right)} \left({x}\right)\:\:{and}\:{f}^{\left({n}\right)} \left(\mathrm{0}\right) \\ $$$$\left.\mathrm{2}\right){developp}\:{f}\:{at}\:{integr}\:{serie} \\ $$$$\left.\mathrm{3}\right)\:{calculate}\:\int_{\mathrm{0}} ^{\mathrm{1}} {f}\left({x}\right){dx} \\ $$

Question Number 68237 Answers: 0 Comments: 1

1. find the equation of the line making an angle of 135^(° ) with O_(x ) and passing through thepoints?(−2,5) 2. find the slope of the line through the points (5,3)and (7,2). find (i) the perpendicular form (ii) find the intercept form of its equation. 3. Determine the gradient of the straight line graph passing through the co-ordinates: (i) (2,7) and (−3,4) (ii) (((1 )/4), ((-3)/4)) and (((-1)/2), (5/8)).

$$\mathrm{1}.\:{find}\:{the}\:{equation}\:{of}\:{the}\:{line}\:{making}\:{an}\:{angle}\:{of}\:\mathrm{135}^{°\:} {with}\:{O}_{{x}\:} \:{and}\:{passing}\:{through}\:{thepoints}?\left(−\mathrm{2},\mathrm{5}\right) \\ $$$$\mathrm{2}.\:{find}\:{the}\:{slope}\:{of}\:{the}\:{line}\:{through}\:{the}\:{points}\:\left(\mathrm{5},\mathrm{3}\right){and}\:\left(\mathrm{7},\mathrm{2}\right).\:{find}\:\left({i}\right)\:{the}\:{perpendicular}\:{form}\:\left({ii}\right)\:{find}\:{the}\:{intercept}\:{form}\:{of}\:{its}\:{equation}. \\ $$$$\mathrm{3}.\:{Determine}\:{the}\:{gradient}\:{of}\:{the}\:{straight}\:{line}\:{graph}\:{passing}\:{through}\:{the}\:{co}-{ordinates}: \\ $$$$\left({i}\right)\:\left(\mathrm{2},\mathrm{7}\right)\:{and}\:\left(−\mathrm{3},\mathrm{4}\right) \\ $$$$\left({ii}\right)\:\left(\frac{\mathrm{1}\:}{\mathrm{4}},\:\frac{-\mathrm{3}}{\mathrm{4}}\right)\:{and}\:\left(\frac{-\mathrm{1}}{\mathrm{2}},\:\frac{\mathrm{5}}{\mathrm{8}}\right). \\ $$

Question Number 68236 Answers: 0 Comments: 0

Question Number 68235 Answers: 0 Comments: 0

Question Number 68234 Answers: 1 Comments: 5

Question Number 68232 Answers: 0 Comments: 0

Question Number 68222 Answers: 1 Comments: 5

Sketch the shear and moment diagrams of a simply supported beam of 6m.The load on the beam consists of UDL of 15KN/m over the left half of the span.

$${Sketch}\:{the}\:{shear}\:{and}\:{moment}\:{diagrams} \\ $$$${of}\:{a}\:{simply}\:{supported}\:{beam}\:{of}\:\mathrm{6}{m}.{The} \\ $$$${load}\:{on}\:{the}\:{beam}\:{consists}\:{of}\:{UDL}\:{of} \\ $$$$\mathrm{15}{KN}/{m}\:{over}\:{the}\:{left}\:{half}\:{of}\:{the}\:{span}. \\ $$$$ \\ $$

Question Number 68220 Answers: 0 Comments: 0

Let consider (a_n )_n and (u_n )_n two reals sequence defined such as a_0 =1 , ∀ n>1 a_(n+1) =Σ_(p=0) ^n a_p a_(n−p) and Σ_(p=0) ^n a_p u_(n−p) =0 Part1 1)Express ∀ n >1 a_n in terms of n 2) Find the largest domain of convergence of the integer serie {a_n x^n } 3)Determinate ∀ x∈D the sum f(x) of {a_n x^n } 4)Find the radius of convergence of the serie {u_n x^n } 5) Give the relation that between the sum S(x) of the second serie and (x/(f(x))) 6) Can you developp in integer serie g(x)=((πx)/(tan(πx))) Part2 Now do the part 1 but in the order 2)−1)−3)−4)−5)−6)

$$\:\:\:{Let}\:{consider}\:\left({a}_{{n}} \right)_{{n}} \:{and}\:\left({u}_{{n}} \right)_{{n}} \:{two}\:{reals}\:\:{sequence}\:\: \\ $$$${defined}\:{such}\:{as}\:\:\:{a}_{\mathrm{0}} =\mathrm{1}\:,\:\forall\:{n}>\mathrm{1}\:\:{a}_{{n}+\mathrm{1}} =\underset{{p}=\mathrm{0}} {\overset{{n}} {\sum}}{a}_{{p}} {a}_{{n}−{p}} \:\:\:{and}\:\:\underset{{p}=\mathrm{0}} {\overset{{n}} {\sum}}{a}_{{p}} {u}_{{n}−{p}} =\mathrm{0} \\ $$$${Part}\mathrm{1} \\ $$$$\left.\mathrm{1}\right){Express}\:\:\forall\:{n}\:>\mathrm{1}\:\:\:{a}_{{n}} \:{in}\:{terms}\:{of}\:{n} \\ $$$$\left.\mathrm{2}\right)\:{Find}\:{the}\:{largest}\:{domain}\:{of}\:{convergence}\:{of}\:{the}\:{integer}\:{serie}\:\left\{{a}_{{n}} {x}^{{n}} \right\} \\ $$$$\left.\mathrm{3}\right){Determinate}\:\forall\:{x}\in{D}\:{the}\:{sum}\:{f}\left({x}\right)\:{of}\:\left\{{a}_{{n}} {x}^{{n}} \right\} \\ $$$$\left.\mathrm{4}\right){Find}\:{the}\:{radius}\:{of}\:{convergence}\:{of}\:{the}\:{serie}\:\left\{{u}_{{n}} {x}^{{n}} \right\}\: \\ $$$$\left.\mathrm{5}\right)\:{Give}\:{the}\:{relation}\:{that}\:{between}\:{the}\:{sum}\:{S}\left({x}\right)\:{of}\:{the}\:{second}\:{serie}\:{and}\:\frac{{x}}{{f}\left({x}\right)}\: \\ $$$$\left.\mathrm{6}\right)\:{Can}\:{you}\:{developp}\:{in}\:{integer}\:{serie}\:\:{g}\left({x}\right)=\frac{\pi{x}}{{tan}\left(\pi{x}\right)} \\ $$$${Part}\mathrm{2} \\ $$$$\left.{N}\left.{o}\left.{w}\left.\:\left.{d}\left.{o}\:\:{the}\:{part}\:\mathrm{1}\:\:\:{but}\:{in}\:{the}\:{order}\:\:\mathrm{2}\right)−\mathrm{1}\right)−\mathrm{3}\right)−\mathrm{4}\right)−\mathrm{5}\right)−\mathrm{6}\right) \\ $$

Question Number 68219 Answers: 1 Comments: 0

Question Number 68212 Answers: 3 Comments: 0

Question Number 68210 Answers: 2 Comments: 0

Question Number 68209 Answers: 0 Comments: 0

Question Number 68207 Answers: 1 Comments: 1

solve for x∈C sin x=z (z=a+bi=re^(iθ) )

$${solve}\:{for}\:{x}\in\mathbb{C} \\ $$$$\mathrm{sin}\:{x}={z}\:\:\:\left({z}={a}+{bi}={re}^{{i}\theta} \right) \\ $$

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