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Question Number 76721 Answers: 2 Comments: 0

what is the mean value of (1/(1+4x^(2 ) )) for 0≤x≤(1/2)

$$\mathrm{what}\:\mathrm{is}\:\mathrm{the}\:\mathrm{mean}\:\mathrm{value}\:\mathrm{of}\:\frac{\mathrm{1}}{\mathrm{1}+\mathrm{4x}^{\mathrm{2}\:} }\:\:\mathrm{for}\:\:\mathrm{0}\leqslant\mathrm{x}\leqslant\frac{\mathrm{1}}{\mathrm{2}} \\ $$

Question Number 76718 Answers: 0 Comments: 0

For n ∈ N prove by mathematical induction that cos α+cos (α+β)+cos [α+(n−1)β]+...cos [α+(n−1)β]= ((cos [α+(((n−1)/2))β]sin ((nβ)/2))/(sin (n/2)))

$${For}\:{n}\:\in\:{N}\:{prove}\:{by}\:{mathematical} \\ $$$${induction}\:{that} \\ $$$$\mathrm{cos}\:\alpha+\mathrm{cos}\:\left(\alpha+\beta\right)+\mathrm{cos}\:\left[\alpha+\left({n}−\mathrm{1}\right)\beta\right]+...\mathrm{cos}\:\left[\alpha+\left({n}−\mathrm{1}\right)\beta\right]= \\ $$$$\frac{\mathrm{cos}\:\left[\alpha+\left(\frac{{n}−\mathrm{1}}{\mathrm{2}}\right)\beta\right]\mathrm{sin}\:\frac{{n}\beta}{\mathrm{2}}}{\mathrm{sin}\:\frac{{n}}{\mathrm{2}}} \\ $$

Question Number 76717 Answers: 1 Comments: 0

A triangle is formed by the three straight line y=m_1 x+(a/m_1 ) y=m_2 x+(a/m_2 ) y=m_3 x+(a/m_3 ) prove that its orthocenter always lies on the line x+a=0

$${A}\:{triangle}\:{is}\:{formed}\:{by} \\ $$$${the}\:{three}\:{straight}\:{line} \\ $$$${y}={m}_{\mathrm{1}} {x}+\frac{{a}}{{m}_{\mathrm{1}} } \\ $$$${y}={m}_{\mathrm{2}} {x}+\frac{{a}}{{m}_{\mathrm{2}} } \\ $$$${y}={m}_{\mathrm{3}} {x}+\frac{{a}}{{m}_{\mathrm{3}} } \\ $$$${prove}\:{that}\:{its}\:{orthocenter} \\ $$$${always}\:{lies}\:{on}\:{the}\:{line} \\ $$$${x}+{a}=\mathrm{0} \\ $$$$ \\ $$

Question Number 76716 Answers: 1 Comments: 4

prove that ∫_0 ^π ((xsin x)/(1+cos^2 x))=(π^2 /4)

$${prove}\:{that} \\ $$$$\int_{\mathrm{0}} ^{\pi} \frac{{x}\mathrm{sin}\:{x}}{\mathrm{1}+\mathrm{cos}\:^{\mathrm{2}} {x}}=\frac{\pi^{\mathrm{2}} }{\mathrm{4}} \\ $$

Question Number 76715 Answers: 0 Comments: 3

If u=arcsin (x/y)+arctan (y/x) show that x(∂u/dx)+y(∂u/dy)=0

$${If}\:{u}={arc}\mathrm{sin}\:\frac{{x}}{{y}}+{arc}\mathrm{tan}\:\frac{{y}}{{x}} \\ $$$${show}\:{that}\: \\ $$$${x}\frac{\partial{u}}{{dx}}+{y}\frac{\partial{u}}{{dy}}=\mathrm{0} \\ $$

Question Number 76714 Answers: 1 Comments: 0

If y=(√(tan x+(√(tan x+(√(tan x+....∞)))))) prove that (dy/dx)=((sec^2 x)/(2y−1))

$${If}\:{y}=\sqrt{\mathrm{tan}\:{x}+\sqrt{\mathrm{tan}\:{x}+\sqrt{\mathrm{tan}\:{x}+....\infty}}}\: \\ $$$${prove}\:{that} \\ $$$$\frac{{dy}}{{dx}}=\frac{\mathrm{sec}\:^{\mathrm{2}} {x}}{\mathrm{2}{y}−\mathrm{1}} \\ $$

Question Number 76713 Answers: 0 Comments: 0

If cos y=xcos (a+y),show that (dy/dx)=((cos^2 (a+y))/(sin a))

$${If}\:\mathrm{cos}\:{y}={x}\mathrm{cos}\:\left({a}+{y}\right),{show} \\ $$$${that}\:\frac{{dy}}{{dx}}=\frac{\mathrm{cos}\:^{\mathrm{2}} \left({a}+{y}\right)}{\mathrm{sin}\:{a}} \\ $$

Question Number 76711 Answers: 1 Comments: 0

Question Number 76700 Answers: 1 Comments: 2

Question Number 76696 Answers: 2 Comments: 0

what is the value ln(0).?

$${what}\:{is}\:{the}\:{value}\:{ln}\left(\mathrm{0}\right).? \\ $$

Question Number 76694 Answers: 0 Comments: 0

Question Number 76688 Answers: 1 Comments: 0

calculate ∫ (1/(lnx))+ ln(lnx) dx .

$${calculate}\:\int\:\frac{\mathrm{1}}{{lnx}}+\:{ln}\left({lnx}\right)\:{dx}\:. \\ $$

Question Number 76680 Answers: 0 Comments: 4

prove that: ∫_0 ^1 (1−x^7 )^(1/3) dx=∫_0 ^1 (1−x^3 )^(1/7) dx

$${prove}\:{that}: \\ $$$$\:\int_{\mathrm{0}} ^{\mathrm{1}} \left(\mathrm{1}−{x}^{\mathrm{7}} \right)^{\frac{\mathrm{1}}{\mathrm{3}}} {dx}=\int_{\mathrm{0}} ^{\mathrm{1}} \left(\mathrm{1}−{x}^{\mathrm{3}} \right)^{\frac{\mathrm{1}}{\mathrm{7}}} {dx} \\ $$

Question Number 76669 Answers: 0 Comments: 2

Question Number 76667 Answers: 2 Comments: 1

Question Number 76664 Answers: 0 Comments: 1

∫_0 ^(+∞) ((t ln(t))/((t^2 +1)^2 )) dt

$$\int_{\mathrm{0}} ^{+\infty} \frac{{t}\:{ln}\left({t}\right)}{\left({t}^{\mathrm{2}} +\mathrm{1}\right)^{\mathrm{2}} }\:{dt} \\ $$

Question Number 76663 Answers: 2 Comments: 0

Question Number 76661 Answers: 1 Comments: 0

calculate ∫ ((x^2 −2)/((x^4 +5x^2 +4) arctan (((x^2 +2)/x)))) dx

$${calculate}\:\int\:\frac{{x}^{\mathrm{2}} −\mathrm{2}}{\left({x}^{\mathrm{4}} +\mathrm{5}{x}^{\mathrm{2}} +\mathrm{4}\right)\:{arc}\mathrm{tan}\:\left(\frac{{x}^{\mathrm{2}} +\mathrm{2}}{{x}}\right)}\:{dx}\: \\ $$

Question Number 76657 Answers: 3 Comments: 0

prove that sin^(−1) (tanhx)=tan^(−1) (sinhx)

$${prove}\:{that}\:{sin}^{−\mathrm{1}} \left({tanhx}\right)={tan}^{−\mathrm{1}} \left({sinhx}\right) \\ $$

Question Number 76651 Answers: 1 Comments: 0

if a and b two numbers do not have to be chosen randomly and with returns from the set {1,2,3,4,5}. what probability that (a/b) is an integer ?

$${if}\:{a}\:{and}\:{b}\:{two}\:{numbers}\:{do}\:{not}\:{have}\: \\ $$$${to}\:{be}\:{chosen}\:{randomly}\:{and}\:{with}\: \\ $$$${returns}\:{from}\:{the}\:{set}\:\left\{\mathrm{1},\mathrm{2},\mathrm{3},\mathrm{4},\mathrm{5}\right\}.\:{what}\: \\ $$$${probability}\:{that}\:\frac{{a}}{{b}}\:{is}\:{an}\:{integer}\:? \\ $$

Question Number 76650 Answers: 0 Comments: 0

two dice are tossed together 4 times. the odds appear that the number on the first dice us twice the number on the second dice exactly twice ?

$${two}\:{dice}\:{are}\:{tossed}\:{together}\:\mathrm{4}\:{times}. \\ $$$${the}\:{odds}\:{appear}\:{that}\:{the}\:{number}\: \\ $$$${on}\:{the}\:{first}\:{dice}\:{us}\:{twice}\:{the}\:{number} \\ $$$${on}\:{the}\:{second}\:{dice}\:{exactly}\:{twice}\:? \\ $$

Question Number 76645 Answers: 1 Comments: 1

Question Number 76633 Answers: 2 Comments: 0

calculate cos^6 ((π/8))+cos^6 (((3π)/8))+cos^6 (((5π)/8))+cos^6 (((7π)/8))

$${calculate}\:{cos}^{\mathrm{6}} \left(\frac{\pi}{\mathrm{8}}\right)+{cos}^{\mathrm{6}} \left(\frac{\mathrm{3}\pi}{\mathrm{8}}\right)+{cos}^{\mathrm{6}} \left(\frac{\mathrm{5}\pi}{\mathrm{8}}\right)+{cos}^{\mathrm{6}} \left(\frac{\mathrm{7}\pi}{\mathrm{8}}\right) \\ $$

Question Number 76630 Answers: 1 Comments: 0

two sequences , (u_n ) and (v_n ), for n∈N is defined as: { ((u_0 =3)),((u_(n+1) = (1/2)(u_n + v_n ) )) :}and { ((v_0 = 4)),((v_(n+1) = (1/2)(u_(n+1) + v_n ))) :} a) calculate u_1 ,v_1 ,u_2 and v_2 b) Another sequence (w_n ), is defined by w_n = v_n − u_n , ∀ n∈N show that w_n is a convegent geometric sequence. c) Express w_n as a function of n and obtain its limits. d) Study the sense of variation(monotony) of (u_n ) and (v_n ) what can you deduce? e) Consider another sequence t_n defined by t_n = ((u_n + 2v_n )/3) , ∀ n ∈ N show that t_n is a constant sequence f) hence obtain the limit of the sequences (u_n ) and (v_n )

$$\mathrm{two}\:\mathrm{sequences}\:,\:\left({u}_{{n}} \right)\:{and}\:\left({v}_{{n}} \right),\:\mathrm{for}\:{n}\in\mathbb{N}\:\mathrm{is}\:\mathrm{defined}\:\mathrm{as}: \\ $$$$\begin{cases}{{u}_{\mathrm{0}} \:=\mathrm{3}}\\{{u}_{{n}+\mathrm{1}} =\:\frac{\mathrm{1}}{\mathrm{2}}\left({u}_{{n}} \:+\:{v}_{{n}} \right)\:\:}\end{cases}\mathrm{and}\:\begin{cases}{{v}_{\mathrm{0}} =\:\mathrm{4}}\\{{v}_{{n}+\mathrm{1}} =\:\frac{\mathrm{1}}{\mathrm{2}}\left({u}_{{n}+\mathrm{1}} \:+\:{v}_{{n}} \right)}\end{cases} \\ $$$$\left.{a}\right)\:\mathrm{calculate}\:{u}_{\mathrm{1}} ,{v}_{\mathrm{1}} ,{u}_{\mathrm{2}} \:\mathrm{and}\:{v}_{\mathrm{2}} \\ $$$$\left.{b}\right)\:\mathrm{Another}\:\mathrm{sequence}\:\left({w}_{{n}} \right),\:\mathrm{is}\:\mathrm{defined}\:\mathrm{by}\: \\ $$$$\:{w}_{{n}} \:=\:{v}_{{n}} \:−\:{u}_{{n}} \:,\:\forall\:{n}\in\mathbb{N} \\ $$$$\mathrm{show}\:\mathrm{that}\:{w}_{{n}} \:\mathrm{is}\:\mathrm{a}\:\mathrm{convegent}\:\mathrm{geometric}\:\mathrm{sequence}. \\ $$$$\left.\mathrm{c}\right)\:\mathrm{Express}\:{w}_{{n}} \:\mathrm{as}\:\mathrm{a}\:\mathrm{function}\:\mathrm{of}\:{n}\:\mathrm{and}\:\mathrm{obtain}\:\mathrm{its}\:\mathrm{limits}. \\ $$$$\left.\mathrm{d}\right)\:\mathrm{Study}\:\mathrm{the}\:\mathrm{sense}\:\mathrm{of}\:\mathrm{variation}\left(\mathrm{monotony}\right)\:\:\mathrm{of}\:\left({u}_{{n}} \right)\:\mathrm{and}\:\left({v}_{{n}} \right) \\ $$$$\mathrm{what}\:\mathrm{can}\:\mathrm{you}\:\mathrm{deduce}? \\ $$$$\left.\mathrm{e}\right)\:\mathrm{Consider}\:\mathrm{another}\:\mathrm{sequence}\:{t}_{{n}} \:\mathrm{defined}\:\mathrm{by} \\ $$$$\:\:\:\:\:\:{t}_{{n}} \:=\:\frac{{u}_{{n}} \:+\:\mathrm{2}{v}_{{n}} }{\mathrm{3}}\:,\:\forall\:{n}\:\in\:\mathbb{N}\: \\ $$$$\mathrm{show}\:\mathrm{that}\:{t}_{{n}} \:\mathrm{is}\:\mathrm{a}\:\mathrm{constant}\:\mathrm{sequence} \\ $$$$\left.\mathrm{f}\right)\:\mathrm{hence}\:\mathrm{obtain}\:\mathrm{the}\:\mathrm{limit}\:\mathrm{of}\:\mathrm{the}\:\mathrm{sequences}\:\:\left({u}_{{n}} \right)\:\mathrm{and}\:\left({v}_{{n}} \right) \\ $$

Question Number 76622 Answers: 2 Comments: 0

Find (a.b.c) for equation acos 2x+bsin^2 x+c=0 is satisfied by every x

$$\mathrm{Find}\:\left(\mathrm{a}.\mathrm{b}.\mathrm{c}\right)\:\mathrm{for}\:\mathrm{equation}\:\mathrm{acos}\:\mathrm{2x}+\mathrm{bsin}\:^{\mathrm{2}} \mathrm{x}+\mathrm{c}=\mathrm{0}\:\mathrm{is}\:\mathrm{satisfied}\:\mathrm{by}\:\mathrm{every}\:\mathrm{x} \\ $$

Question Number 76615 Answers: 2 Comments: 0

∫sec^3 x dx

$$\int{sec}^{\mathrm{3}} {x}\:{dx} \\ $$

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