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Question Number 88547 Answers: 1 Comments: 0

prove for (0<a<2) ∫_0 ^( ∞) ((x^(a−1) dx)/(1+x+x^2 )) = ((2π)/(√3))cos (((2πa+π)/6))cosec πa .

$${prove}\:{for}\:\left(\mathrm{0}<{a}<\mathrm{2}\right) \\ $$$$\int_{\mathrm{0}} ^{\:\infty} \frac{{x}^{{a}−\mathrm{1}} {dx}}{\mathrm{1}+{x}+{x}^{\mathrm{2}} }\:=\:\frac{\mathrm{2}\pi}{\sqrt{\mathrm{3}}}\mathrm{cos}\:\left(\frac{\mathrm{2}\pi{a}+\pi}{\mathrm{6}}\right)\mathrm{cosec}\:\pi{a}\:. \\ $$

Question Number 88541 Answers: 1 Comments: 5

Question Number 88525 Answers: 1 Comments: 6

Question Number 88507 Answers: 1 Comments: 1

cos^2 12^o +cos^2 24^o +cos^2 48^o +cos^2 84^o

$$\mathrm{cos}\:^{\mathrm{2}} \mathrm{12}^{\mathrm{o}} +\mathrm{cos}\:^{\mathrm{2}} \mathrm{24}^{\mathrm{o}} +\mathrm{cos}\:^{\mathrm{2}} \mathrm{48}^{\mathrm{o}} +\mathrm{cos}\:^{\mathrm{2}} \mathrm{84}^{\mathrm{o}} \\ $$

Question Number 88503 Answers: 0 Comments: 3

(1+cos (π/8))(1+cos ((3π)/8))(1+cos ((5π)/8))(1+cos ((7π)/8))

$$\left(\mathrm{1}+\mathrm{cos}\:\frac{\pi}{\mathrm{8}}\right)\left(\mathrm{1}+\mathrm{cos}\:\frac{\mathrm{3}\pi}{\mathrm{8}}\right)\left(\mathrm{1}+\mathrm{cos}\:\frac{\mathrm{5}\pi}{\mathrm{8}}\right)\left(\mathrm{1}+\mathrm{cos}\:\frac{\mathrm{7}\pi}{\mathrm{8}}\right) \\ $$$$ \\ $$

Question Number 88494 Answers: 0 Comments: 2

∫ (√(cos(x))) dx

$$\int\:\sqrt{\mathrm{cos}\left(\mathrm{x}\right)}\:\:\mathrm{dx} \\ $$

Question Number 88492 Answers: 0 Comments: 4

y′′ −4y′+5y = 1+8cos x+e^(2x)

$$\mathrm{y}''\:−\mathrm{4y}'+\mathrm{5y}\:=\:\mathrm{1}+\mathrm{8cos}\:\mathrm{x}+\mathrm{e}^{\mathrm{2x}} \\ $$

Question Number 88491 Answers: 1 Comments: 0

solve cos(x)=k

$$\boldsymbol{{solve}} \\ $$$${cos}\left({x}\right)={k} \\ $$

Question Number 88490 Answers: 0 Comments: 4

∫_1 ^∞ (x^4 /4^x )dx=?

$$\int_{\mathrm{1}} ^{\infty} \:\frac{{x}^{\mathrm{4}} }{\mathrm{4}^{{x}} }{dx}=? \\ $$

Question Number 88487 Answers: 0 Comments: 7

Question Number 88479 Answers: 0 Comments: 0

Consider the transformation f of the plane with all points M wity affix z mapped to the point M ′ with affix z ′ such that z ′=−((√3)+i)z−1+i(1+(√3)) 1) Given M_0 the point z_0 =((√3)/4)+(3/4)i calculate AM_0 and deduce the angle in radians (Taking A as the center of the transformation) 2) Consider the progression with points(M_n )_(n≥0) defined by f(M_n )=M_(n+1) a∙ Show by recurrence that ∀n∈N z_n =2^n e^(ln((7π)/6)) (z_(0 ) −i) Find AM_n then determine the smallest natural number, n, such that AM_n ≥10^2

$${Consider}\:{the}\:{transformation}\:\boldsymbol{{f}}\:{of}\:{the}\:{plane}\:{with}\:{all}\:{points} \\ $$$$\boldsymbol{{M}}\:{wity}\:{affix}\:\boldsymbol{{z}}\:{mapped}\:{to}\:{the}\:{point}\:\boldsymbol{{M}}\:'\:{with}\:{affix}\:\boldsymbol{{z}}\:' \\ $$$${such}\:{that}\:\boldsymbol{{z}}\:'=−\left(\sqrt{\mathrm{3}}+{i}\right){z}−\mathrm{1}+{i}\left(\mathrm{1}+\sqrt{\mathrm{3}}\right) \\ $$$$\left.\mathrm{1}\right)\:{Given}\:\boldsymbol{{M}}_{\mathrm{0}} \:{the}\:{point}\:\boldsymbol{{z}}_{\mathrm{0}} =\frac{\sqrt{\mathrm{3}}}{\mathrm{4}}+\frac{\mathrm{3}}{\mathrm{4}}{i} \\ $$$${calculate}\:\boldsymbol{{AM}}_{\mathrm{0}} \:{and}\:{deduce}\:{the}\:{angle}\:{in}\:{radians} \\ $$$$\left({Taking}\:\boldsymbol{{A}}\:{as}\:{the}\:{center}\:{of}\:{the}\:{transformation}\right) \\ $$$$\left.\mathrm{2}\right)\:{Consider}\:{the}\:{progression}\:{with}\:{points}\left(\boldsymbol{{M}}_{\boldsymbol{{n}}} \right)_{\boldsymbol{{n}}\geqslant\mathrm{0}} \:{defined}\:{by} \\ $$$${f}\left({M}_{{n}} \right)={M}_{{n}+\mathrm{1}} \\ $$$${a}\centerdot\:{Show}\:{by}\:{recurrence}\:{that}\:\forall{n}\in\mathbb{N}\:\boldsymbol{{z}}_{\boldsymbol{{n}}} =\mathrm{2}^{{n}} {e}^{{ln}\frac{\mathrm{7}\pi}{\mathrm{6}}} \:\left({z}_{\mathrm{0}\:} −{i}\right) \\ $$$${Find}\:{AM}_{{n}} \:{then}\:{determine}\:{the}\:{smallest}\:{natural}\:{number},\:{n},\:{such}\:{that} \\ $$$${AM}_{{n}} \geqslant\mathrm{10}^{\mathrm{2}} \\ $$

Question Number 88473 Answers: 0 Comments: 2

Question Number 88471 Answers: 1 Comments: 0

Question Number 88461 Answers: 0 Comments: 2

Question Number 88458 Answers: 0 Comments: 0

Using the principle of mathematical induction to prove that a_1 , a_2 , ... , a_n , ((a_1 + a_2 + ... + a_n )/n) ≥ ((a_1 , a_2 , ... , a_n ))^(1/n)

$$\mathrm{Using}\:\mathrm{the}\:\mathrm{principle}\:\mathrm{of}\:\mathrm{mathematical}\:\mathrm{induction}\:\mathrm{to}\:\mathrm{prove} \\ $$$$\mathrm{that}\:\:\:\mathrm{a}_{\mathrm{1}} \:,\:\:\:\mathrm{a}_{\mathrm{2}} \:,\:\:...\:,\:\mathrm{a}_{\mathrm{n}} \:,\:\:\frac{\mathrm{a}_{\mathrm{1}} \:+\:\mathrm{a}_{\mathrm{2}} \:+\:...\:+\:\mathrm{a}_{\mathrm{n}} }{\mathrm{n}}\:\:\:\:\geqslant\:\:\:\sqrt[{\mathrm{n}}]{\mathrm{a}_{\mathrm{1}} \:,\:\:\mathrm{a}_{\mathrm{2}} \:,\:\:...\:,\:\mathrm{a}_{\mathrm{n}} } \\ $$

Question Number 88456 Answers: 1 Comments: 4

Question Number 88462 Answers: 0 Comments: 3

Question Number 88440 Answers: 2 Comments: 2

prove Σ_(k=1) ^∞ ((4^k −3^k )/(12^k ))=(1/6)

$${prove}\:\:\underset{{k}=\mathrm{1}} {\overset{\infty} {\sum}}\frac{\mathrm{4}^{{k}} −\mathrm{3}^{{k}} }{\mathrm{12}^{{k}} }=\frac{\mathrm{1}}{\mathrm{6}} \\ $$

Question Number 88439 Answers: 0 Comments: 4

kofi have three books on his desk, they are mathematics , biology and physics. Ama also have three books on her desk namely physics ,mathematics and chemistry. A thief picked one book from each of their desk. what is the probability of picking mathematics book

$$\mathrm{kofi}\:\mathrm{have}\:\mathrm{three}\:\mathrm{books}\:\mathrm{on}\:\mathrm{his}\:\mathrm{desk},\:\mathrm{they} \\ $$$$\mathrm{are}\:\mathrm{mathematics}\:,\:\mathrm{biology}\:\mathrm{and}\:\mathrm{physics}. \\ $$$$\mathrm{Ama}\:\mathrm{also}\:\mathrm{have}\:\mathrm{three}\:\mathrm{books}\:\mathrm{on}\: \\ $$$$\mathrm{her}\:\mathrm{desk}\:\mathrm{namely}\:\mathrm{physics}\:,\mathrm{mathematics} \\ $$$$\mathrm{and}\:\mathrm{chemistry}.\:\mathrm{A}\:\mathrm{thief}\:\mathrm{picked}\:\mathrm{one}\:\mathrm{book} \\ $$$$\mathrm{from}\:\mathrm{each}\:\mathrm{of}\:\mathrm{their}\:\mathrm{desk}.\:\mathrm{what}\:\mathrm{is}\:\mathrm{the} \\ $$$$\mathrm{probability}\:\mathrm{of}\:\mathrm{picking}\:\mathrm{mathematics}\: \\ $$$$\mathrm{book} \\ $$

Question Number 88438 Answers: 2 Comments: 0

∫((x^5 +1)/(x^5 −1))dx

$$\int\frac{{x}^{\mathrm{5}} +\mathrm{1}}{{x}^{\mathrm{5}} −\mathrm{1}}{dx} \\ $$

Question Number 88436 Answers: 0 Comments: 0

show that (for e>1) the equation of a hyperbola with focus (±ae,0) and directrix x = (a/e) is (x^2 /a^2 ) − (y^2 /b^2 ) hence find an equation for the eccencitrity of the hyperbola

$$\:\mathrm{show}\:\mathrm{that}\:\left(\mathrm{for}\:{e}>\mathrm{1}\right)\:\mathrm{the}\:\mathrm{equation}\:\mathrm{of}\:\mathrm{a}\:\mathrm{hyperbola}\:\mathrm{with}\: \\ $$$$\mathrm{focus}\:\:\left(\pm{ae},\mathrm{0}\right)\:\mathrm{and}\:\mathrm{directrix}\:\:{x}\:=\:\frac{{a}}{{e}}\:\mathrm{is}\:\frac{{x}^{\mathrm{2}} }{{a}^{\mathrm{2}} }\:−\:\frac{{y}^{\mathrm{2}} }{{b}^{\mathrm{2}} } \\ $$$$\:\:\:\mathrm{hence}\:\mathrm{find}\:\mathrm{an}\:\mathrm{equation}\:\mathrm{for}\:\mathrm{the}\:\mathrm{eccencitrity}\:\mathrm{of}\: \\ $$$$\mathrm{the}\:\mathrm{hyperbola} \\ $$

Question Number 88435 Answers: 0 Comments: 0

Question Number 88434 Answers: 0 Comments: 2

find the partial derivatives of first and second order for the function f(x,y) = x^3 y + 3xy + y^4

$$\mathrm{find}\:\mathrm{the}\:\mathrm{partial}\:\mathrm{derivatives}\:\mathrm{of}\:\mathrm{first}\:\mathrm{and}\:\mathrm{second}\:\mathrm{order} \\ $$$$\mathrm{for}\:\mathrm{the}\:\mathrm{function} \\ $$$$\:{f}\left({x},{y}\right)\:=\:{x}^{\mathrm{3}} {y}\:+\:\mathrm{3}{xy}\:+\:{y}^{\mathrm{4}} \\ $$

Question Number 88430 Answers: 1 Comments: 1

Question Number 88429 Answers: 0 Comments: 0

show that ∫_(0 ) ^1 ln(x) sin^(−1) (√x) dx= (π/2)(ln(2)−1)

$${show}\:{that} \\ $$$$\int_{\mathrm{0}\:} ^{\mathrm{1}} {ln}\left({x}\right)\:{sin}^{−\mathrm{1}} \sqrt{{x}}\:{dx}=\:\frac{\pi}{\mathrm{2}}\left({ln}\left(\mathrm{2}\right)−\mathrm{1}\right) \\ $$

Question Number 88424 Answers: 0 Comments: 1

calculate U_n =∫_0 ^∞ ((arctan(n^2 x)−arctan(nx))/x)dx and xetermine nature of the serie Σ U_n

$${calculate}\:{U}_{{n}} =\int_{\mathrm{0}} ^{\infty} \:\:\frac{{arctan}\left({n}^{\mathrm{2}} {x}\right)−{arctan}\left({nx}\right)}{{x}}{dx} \\ $$$${and}\:{xetermine}\:{nature}\:{of}\:{the}\:{serie}\:\Sigma\:{U}_{{n}} \\ $$

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