Question and Answers Forum

AllQuestion and Answers: Page 1367

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

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

Question Number 76588 Answers: 2 Comments: 0

Given three events A,B, and C sucb that P(A) = P(C) , P(A ∩ C) = (1/(10)) , P(A ∪ C)=(1/2) P(C∣B) = (2/7) , P(B∪C) = (4/5) find a) P(A) b) P(B)

Question Number 76587 Answers: 1 Comments: 3

Question Number 76586 Answers: 1 Comments: 7

Question Number 76585 Answers: 4 Comments: 1

Question Number 76580 Answers: 0 Comments: 5

how many divisors does 180045 has?

$$\mathrm{how}\:\mathrm{many}\:\mathrm{divisors}\:\mathrm{does}\:\mathrm{180045}\:\mathrm{has}? \\ $$

Question Number 76579 Answers: 1 Comments: 0

A uniform ladder of weight W and length 2a rest in limiting equilibrium with one end on a rough horizontal ground and the other end on a rough vertical wall. The coefficient of friction between the ladder and the ground and between the ladder and the wall are respectively μ and λ . If the ladder makes an angle θ with the ground where tan θ = (5/(12)), a) show that 5μ + 6λμ − 6 = 0. b) find the value of λ and μ given that λμ =(1/2).

$$\mathrm{A}\:\mathrm{uniform}\:\mathrm{ladder}\:\mathrm{of}\:\mathrm{weight}\:{W}\:\mathrm{and}\:\mathrm{length} \\ $$$$\mathrm{2}{a}\:\mathrm{rest}\:\mathrm{in}\:\mathrm{limiting}\:\mathrm{equilibrium}\:\mathrm{with}\:\mathrm{one}\: \\ $$$$\mathrm{end}\:\mathrm{on}\:\mathrm{a}\:\mathrm{rough}\:\mathrm{horizontal}\:\mathrm{ground}\:\mathrm{and}\:\mathrm{the} \\ $$$$\mathrm{other}\:\mathrm{end}\:\mathrm{on}\:\mathrm{a}\:\mathrm{rough}\:\mathrm{vertical}\:\mathrm{wall}. \\ $$$$\mathrm{The}\:\mathrm{coefficient}\:\mathrm{of}\:\mathrm{friction}\:\mathrm{between}\:\mathrm{the}\:\mathrm{ladder} \\ $$$$\mathrm{and}\:\mathrm{the}\:\mathrm{ground}\:\mathrm{and}\:\mathrm{between}\:\mathrm{the}\:\mathrm{ladder}\:\mathrm{and} \\ $$$$\mathrm{the}\:\mathrm{wall}\:\mathrm{are}\:\mathrm{respectively}\:\mu\:\mathrm{and}\:\lambda\:.\:\mathrm{If}\:\mathrm{the}\:\mathrm{ladder} \\ $$$$\mathrm{makes}\:\mathrm{an}\:\mathrm{angle}\:\theta\:\mathrm{with}\:\mathrm{the}\:\mathrm{ground}\:\mathrm{where}\:\mathrm{tan}\:\theta\:=\:\frac{\mathrm{5}}{\mathrm{12}}, \\ $$$$\left.\mathrm{a}\right)\:\mathrm{show}\:\mathrm{that}\:\mathrm{5}\mu\:+\:\mathrm{6}\lambda\mu\:−\:\mathrm{6}\:=\:\mathrm{0}. \\ $$$$\left.\mathrm{b}\right)\:\mathrm{find}\:\mathrm{the}\:\mathrm{value}\:\mathrm{of}\:\lambda\:\mathrm{and}\:\mu\:\mathrm{given}\:\mathrm{that}\:\lambda\mu\:=\frac{\mathrm{1}}{\mathrm{2}}.\: \\ $$

Question Number 76577 Answers: 1 Comments: 0

given a sequence defined by {((3n)/(2n+ 5))}_(n=1) ^∞ , does this sequence converge or diverge, explain

$$\:{given}\:{a}\:{sequence}\:{defined}\:{by}\:\:\left\{\frac{\mathrm{3}{n}}{\mathrm{2}{n}+\:\mathrm{5}}\right\}_{{n}=\mathrm{1}} ^{\infty} ,\:{does}\:{this}\: \\ $$$${sequence}\:{converge}\:{or}\:{diverge},\:{explain} \\ $$

Question Number 76576 Answers: 1 Comments: 0

Compute the Gcd (greatest common divisor) of 45 and 1287

$${Compute}\:{the}\:{Gcd}\:\left({greatest}\:{common}\:{divisor}\right) \\ $$$${of}\:\:\mathrm{45}\:{and}\:\mathrm{1287}\: \\ $$

Question Number 76575 Answers: 1 Comments: 0

define the concept of a contingency Which of the following is a contingency and which is a tautology 1) (P ⇒∼P) ∨ Q 2) (P ⇒ ∼P) ⇒ Q

$${define}\:{the}\:{concept}\:{of}\:{a}\:\boldsymbol{{contingency}} \\ $$$${Which}\:{of}\:{the}\:{following}\:{is}\:{a}\:{contingency}\: \\ $$$${and}\:{which}\:{is}\:{a}\:{tautology} \\ $$$$\left.\mathrm{1}\right)\:\left({P}\:\Rightarrow\sim{P}\right)\:\vee\:{Q}\:\: \\ $$$$\left.\mathrm{2}\right)\:\left({P}\:\Rightarrow\:\sim{P}\right)\:\Rightarrow\:{Q} \\ $$

Question Number 76574 Answers: 0 Comments: 0