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Question Number 145406 Answers: 2 Comments: 0

if log_a c+log_c b=2 ; log_b c+log_a c=0 find (1/(log_a b)) + (1/(log_b c)) + (1/(log_c a)) = ?

$${if}\:\:\boldsymbol{{log}}_{\boldsymbol{{a}}} \boldsymbol{{c}}+\boldsymbol{{log}}_{\boldsymbol{{c}}} \boldsymbol{{b}}=\mathrm{2}\:;\:\boldsymbol{{log}}_{\boldsymbol{{b}}} \boldsymbol{{c}}+\boldsymbol{{log}}_{\boldsymbol{{a}}} \boldsymbol{{c}}=\mathrm{0} \\ $$$${find}\:\:\frac{\mathrm{1}}{\boldsymbol{{log}}_{\boldsymbol{{a}}} \boldsymbol{{b}}}\:+\:\frac{\mathrm{1}}{\boldsymbol{{log}}_{\boldsymbol{{b}}} \boldsymbol{{c}}}\:+\:\frac{\mathrm{1}}{\boldsymbol{{log}}_{\boldsymbol{{c}}} \boldsymbol{{a}}}\:=\:? \\ $$

Question Number 145399 Answers: 1 Comments: 0

Prove that lim_(n→+∞) ∫^( n) _( 0) (t^n /(n!)) e^(−t) dt = (1/2)

$$\mathrm{Prove}\:\mathrm{that}\: \\ $$$$\underset{\boldsymbol{\mathrm{n}}\rightarrow+\infty} {\boldsymbol{\mathrm{lim}}}\:\:\underset{\:\mathrm{0}} {\int}^{\:\boldsymbol{\mathrm{n}}} \:\frac{\boldsymbol{\mathrm{t}}^{\boldsymbol{\mathrm{n}}} }{\boldsymbol{\mathrm{n}}!}\:\boldsymbol{{e}}^{−\boldsymbol{\mathrm{t}}} \:\boldsymbol{\mathrm{dt}}\:=\:\frac{\mathrm{1}}{\mathrm{2}} \\ $$

Question Number 145393 Answers: 2 Comments: 0

The number (2/(13)) expressed as a decimal is 0.153846153846... The 200th and 300th digits are?

$$\mathrm{The}\:\mathrm{number}\:\frac{\mathrm{2}}{\mathrm{13}}\:\mathrm{expressed}\:\mathrm{as}\:\mathrm{a}\:\mathrm{decimal}\:\mathrm{is}\:\mathrm{0}.\mathrm{153846153846}... \\ $$$$\mathrm{The}\:\mathrm{200th}\:\mathrm{and}\:\mathrm{300th}\:\mathrm{digits}\:\mathrm{are}? \\ $$

Question Number 145391 Answers: 2 Comments: 0

Developpement limite^ a l′ordre 2 de g(x)=((√(1+x^2 ))/(1+x+(√(1+x^2 ))))

$$\mathrm{Developpement}\:\mathrm{limit}\acute {\mathrm{e}}\:\mathrm{a}\:\mathrm{l}'\mathrm{ordre}\:\mathrm{2}\:\mathrm{de}\: \\ $$$$\mathrm{g}\left(\mathrm{x}\right)=\frac{\sqrt{\mathrm{1}+\mathrm{x}^{\mathrm{2}} }}{\mathrm{1}+\mathrm{x}+\sqrt{\mathrm{1}+\mathrm{x}^{\mathrm{2}} }} \\ $$

Question Number 145390 Answers: 1 Comments: 0

ϕ(x)=ln(((e^(x+cos(x)) −e)/(x+x^2 ))) montrer que ϕ se prolonge par continuite^ en 0. on note ψ son prolongement, montrer que ψ est de^ rivable en 0.. Ainsi donner une e^ quation de la tangente, position de la courbe par rapport a la tangente, et faire le dessin..

$$\varphi\left(\mathrm{x}\right)=\mathrm{ln}\left(\frac{\mathrm{e}^{\mathrm{x}+\mathrm{cos}\left(\mathrm{x}\right)} −\mathrm{e}}{\mathrm{x}+\mathrm{x}^{\mathrm{2}} }\right) \\ $$$$\mathrm{montrer}\:\mathrm{que}\:\varphi\:\mathrm{se}\:\mathrm{prolonge}\:\mathrm{par}\:\mathrm{continuit}\acute {\mathrm{e}} \\ $$$$\mathrm{en}\:\mathrm{0}.\:\mathrm{on}\:\mathrm{note}\:\psi\:\mathrm{son}\:\mathrm{prolongement},\:\mathrm{montrer} \\ $$$$\mathrm{que}\:\psi\:\mathrm{est}\:\mathrm{d}\acute {\mathrm{e}rivable}\:\mathrm{en}\:\mathrm{0}..\:\:\mathrm{Ainsi}\:\mathrm{donner}\:\mathrm{une} \\ $$$$\acute {\mathrm{e}quation}\:\mathrm{de}\:\mathrm{la}\:\mathrm{tangente},\:\mathrm{position}\:\mathrm{de}\:\mathrm{la}\:\mathrm{courbe} \\ $$$$\mathrm{par}\:\mathrm{rapport}\:\mathrm{a}\:\mathrm{la}\:\mathrm{tangente},\:\mathrm{et}\:\mathrm{faire}\:\mathrm{le}\:\mathrm{dessin}.. \\ $$

Question Number 145385 Answers: 1 Comments: 0

∫_0 ^( (π/2)) ((1+cos (2x))/(sin (2x ))). ln((sec (x)))^(1/3) dx=?

$$\int_{\mathrm{0}} ^{\:\frac{\pi}{\mathrm{2}}} \frac{\mathrm{1}+\mathrm{cos}\:\left(\mathrm{2x}\right)}{\mathrm{sin}\:\left(\mathrm{2x}\:\right)}.\:\mathrm{ln}\sqrt[{\mathrm{3}}]{\mathrm{sec}\:\left(\mathrm{x}\right)}\:\mathrm{dx}=? \\ $$

Question Number 145383 Answers: 2 Comments: 0

if a;b;c∈R^+ find (((abc))^(1/3) + (1/a) + (1/(2b)) + (1/(4c)))_(min) = ?

$${if}\:\:{a};{b};{c}\in\mathbb{R}^{+} \\ $$$${find}\:\:\left(\sqrt[{\mathrm{3}}]{{abc}}\:+\:\frac{\mathrm{1}}{{a}}\:+\:\frac{\mathrm{1}}{\mathrm{2}{b}}\:+\:\frac{\mathrm{1}}{\mathrm{4}{c}}\right)_{\boldsymbol{{min}}} =\:? \\ $$

Question Number 145379 Answers: 1 Comments: 1

Question Number 145378 Answers: 1 Comments: 0

∫_0 ^( ∞) ((√( 1+ x^4 )) −x^( 2) )dx=?

$$ \\ $$$$\:\:\:\int_{\mathrm{0}} ^{\:\infty} \:\left(\sqrt{\:\mathrm{1}+\:\mathrm{x}^{\mathrm{4}} }\:−\mathrm{x}^{\:\mathrm{2}} \:\right)\mathrm{dx}=? \\ $$

Question Number 145373 Answers: 1 Comments: 0

Question Number 145370 Answers: 0 Comments: 0

There are two circles , C of radius 1 and C_r of radius r which intersect on a plain At each of the two intersecting points on the circumferences of C and C_r ,the tangent to C and that to C_r form an angle 120° outside of C and C_r . Fill in the blanks with the answers to the following questions (1) Express the distance d between the centers of C and C_r in terms of r (2) Calculate the value of r at which d in (1) attains the minimum (3) in case(2) express the area of the intersection of C and C_r in terms of the constant π

Question Number 145363 Answers: 3 Comments: 0

Without L′Hopital rule lim_(x→π/4) (((√2) cos x−1)/(cot x−1)) =?

$$\:\mathrm{Without}\:\mathrm{L}'\mathrm{Hopital}\:\mathrm{rule} \\ $$$$\:\underset{{x}\rightarrow\pi/\mathrm{4}} {\mathrm{lim}}\:\frac{\sqrt{\mathrm{2}}\:\mathrm{cos}\:\mathrm{x}−\mathrm{1}}{\mathrm{cot}\:\mathrm{x}−\mathrm{1}}\:=? \\ $$

Question Number 145361 Answers: 0 Comments: 0

∫_0 ^(+∞) ((t^2 +3t+3)/((t+1)^3 )) e^(−t) cos(t) dt

$$\int_{\mathrm{0}} ^{+\infty} \frac{\mathrm{t}^{\mathrm{2}} +\mathrm{3t}+\mathrm{3}}{\left(\mathrm{t}+\mathrm{1}\right)^{\mathrm{3}} }\:\mathrm{e}^{−\mathrm{t}} \mathrm{cos}\left(\mathrm{t}\right)\:\mathrm{dt} \\ $$

Question Number 145359 Answers: 1 Comments: 0

How many digits will there be in 875^(16) ?

$$\mathrm{How}\:\mathrm{many}\:\mathrm{digits}\:\mathrm{will}\:\mathrm{there}\:\mathrm{be} \\ $$$$\mathrm{in}\:\mathrm{875}^{\mathrm{16}} \:? \\ $$

Question Number 145358 Answers: 1 Comments: 0

Evaluate:: ∫_0 ^1 ln(1+x^2 )∙arctan(x)dx=?

$$\mathrm{Evaluate}::\:\:\:\int_{\mathrm{0}} ^{\mathrm{1}} \mathrm{ln}\left(\mathrm{1}+\mathrm{x}^{\mathrm{2}} \right)\centerdot\mathrm{arctan}\left(\mathrm{x}\right)\mathrm{dx}=? \\ $$

Question Number 145345 Answers: 1 Comments: 0

Σ_(n=1) ^∞ (((−1)^n n)/((2n+1)!))=?

$$\underset{\mathrm{n}=\mathrm{1}} {\overset{\infty} {\sum}}\frac{\left(−\mathrm{1}\right)^{\mathrm{n}} \mathrm{n}}{\left(\mathrm{2n}+\mathrm{1}\right)!}=? \\ $$

Question Number 145344 Answers: 2 Comments: 0

z^4 = 49 - 20(√6) ⇒ z=?

$$\boldsymbol{{z}}^{\mathrm{4}} \:=\:\mathrm{49}\:-\:\mathrm{20}\sqrt{\mathrm{6}}\:\:\Rightarrow\:\boldsymbol{{z}}=? \\ $$

Question Number 145339 Answers: 1 Comments: 0

Let f(x)=e^x cos x,Find Σ_(n=0) ^∞ ((f^((n)) (x))/2^n )=?

$$\mathrm{Let}\:\mathrm{f}\left(\mathrm{x}\right)=\mathrm{e}^{\mathrm{x}} \mathrm{cos}\:\mathrm{x},\mathrm{Find}\:\underset{\mathrm{n}=\mathrm{0}} {\overset{\infty} {\sum}}\frac{\mathrm{f}^{\left(\mathrm{n}\right)} \left(\mathrm{x}\right)}{\mathrm{2}^{\mathrm{n}} }=? \\ $$

Question Number 145338 Answers: 1 Comments: 0

de^ rive^ e n-ie^ me de (x^3 /(1+x^6 ))

$$\mathrm{d}\acute {\mathrm{e}riv}\acute {\mathrm{e}e}\:\:\:\mathrm{n}-\mathrm{i}\grave {\mathrm{e}me}\:\:\:\mathrm{de}\:\:\:\frac{\mathrm{x}^{\mathrm{3}} }{\mathrm{1}+\mathrm{x}^{\mathrm{6}} } \\ $$

Question Number 145337 Answers: 1 Comments: 0

The Area of a square,A(t),is increased at a rate equal to its perimeter ,A(t) satisfies the differential equation (dA/dt)= A. 4A B. 2A C.−4(√A) D. 4(√A)

$$\mathrm{The}\:\mathrm{Area}\:\mathrm{of}\:\mathrm{a}\:\mathrm{square},{A}\left({t}\right),\mathrm{is}\:\mathrm{increased}\:\mathrm{at}\:\mathrm{a}\:\mathrm{rate} \\ $$$$\mathrm{equal}\:\mathrm{to}\:\mathrm{its}\:\mathrm{perimeter}\:,{A}\left({t}\right)\:\mathrm{satisfies}\:\mathrm{the}\:\mathrm{differential} \\ $$$$\mathrm{equation}\:\frac{{dA}}{{dt}}= \\ $$$$\mathrm{A}.\:\mathrm{4}{A}\:\:\:\:\:\:\:\:\:\mathrm{B}.\:\mathrm{2}{A}\:\:\:\:\:\:\:\:\:\mathrm{C}.−\mathrm{4}\sqrt{{A}}\:\:\:\:\:\:\:\:\mathrm{D}.\:\mathrm{4}\sqrt{{A}} \\ $$

Question Number 145336 Answers: 0 Comments: 4

Which of the following linear diophantine equations has positive solutions, A. x+ 5y = 7 B. x+5y = 3 B. x + 5y = 2 C. x+ 5y = 1

$$\mathrm{Which}\:\mathrm{of}\:\mathrm{the}\:\mathrm{following}\:\mathrm{linear}\:\mathrm{diophantine}\:\mathrm{equations}\:\mathrm{has} \\ $$$$\mathrm{positive}\:\mathrm{solutions}, \\ $$$$\mathrm{A}.\:{x}+\:\mathrm{5}{y}\:=\:\mathrm{7} \\ $$$$\mathrm{B}.\:{x}+\mathrm{5}{y}\:=\:\mathrm{3} \\ $$$$\mathrm{B}.\:{x}\:+\:\mathrm{5}{y}\:=\:\mathrm{2} \\ $$$$\mathrm{C}.\:{x}+\:\mathrm{5}{y}\:=\:\mathrm{1} \\ $$

Question Number 145324 Answers: 2 Comments: 0

∫ (dx/(1 + x^6 )) = ?

$$\int\:\frac{{dx}}{\mathrm{1}\:+\:{x}^{\mathrm{6}} }\:=\:? \\ $$

Question Number 145319 Answers: 1 Comments: 0

# Calculus# Σ_(n=0) ^∞ (1/(n! + (n + 1 )!)) =?

$$\:\:\:\:\: \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:#\:\:\mathrm{Calculus}# \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\underset{{n}=\mathrm{0}} {\overset{\infty} {\sum}}\:\frac{\mathrm{1}}{{n}!\:+\:\left({n}\:+\:\mathrm{1}\:\right)!}\:=? \\ $$$$ \\ $$

Question Number 145318 Answers: 1 Comments: 0

Let f:[0,1]→R be a differentiable function such that f(f(x))=x for all x∈[0,1] and f(0)=1. If n is a positive integer, evaluate the following integral: ∫_0 ^( 1) (x−f(x))^(2n) dx

$$\mathrm{Let}\:{f}:\left[\mathrm{0},\mathrm{1}\right]\rightarrow\mathbb{R}\:\mathrm{be}\:\mathrm{a}\:\mathrm{differentiable}\:\mathrm{function} \\ $$$$\mathrm{such}\:\mathrm{that}\:{f}\left({f}\left({x}\right)\right)={x}\:\mathrm{for}\:\mathrm{all}\:{x}\in\left[\mathrm{0},\mathrm{1}\right]\:\mathrm{and} \\ $$$${f}\left(\mathrm{0}\right)=\mathrm{1}. \\ $$$$\mathrm{If}\:{n}\:\mathrm{is}\:\mathrm{a}\:\mathrm{positive}\:\mathrm{integer},\:\mathrm{evaluate}\:\mathrm{the} \\ $$$$\mathrm{following}\:\mathrm{integral}:\: \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\int_{\mathrm{0}} ^{\:\mathrm{1}} \left({x}−{f}\left({x}\right)\right)^{\mathrm{2}{n}} \:{dx} \\ $$

Question Number 145316 Answers: 2 Comments: 2

Question Number 145314 Answers: 0 Comments: 2

Let a,b ≥ 0 and (a+1)(b+1) = (a+b)^2 . Prove that (a+b)(√((a+1)^3 +(b+1)^3 )) ≤ (a+1)^2 +(b+1)^2 ≤ (1/2)[(a+1)^3 +(b+1)^3 ]

$$\mathrm{Let}\:{a},{b}\:\geqslant\:\mathrm{0}\:\mathrm{and}\:\left({a}+\mathrm{1}\right)\left({b}+\mathrm{1}\right)\:=\:\left({a}+{b}\right)^{\mathrm{2}} \:.\:\mathrm{Prove}\:\mathrm{that} \\ $$$$\:\:\:\:\:\:\:\left({a}+{b}\right)\sqrt{\left({a}+\mathrm{1}\right)^{\mathrm{3}} +\left({b}+\mathrm{1}\right)^{\mathrm{3}} }\:\leqslant\:\left({a}+\mathrm{1}\right)^{\mathrm{2}} +\left({b}+\mathrm{1}\right)^{\mathrm{2}} \:\leqslant\:\frac{\mathrm{1}}{\mathrm{2}}\left[\left({a}+\mathrm{1}\right)^{\mathrm{3}} +\left({b}+\mathrm{1}\right)^{\mathrm{3}} \right] \\ $$

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