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Question Number 140899 Answers: 0 Comments: 2

Question Number 140896 Answers: 2 Comments: 0

the function f with variable x satisfies the equation x^2 f ′(x) +2x f(x) = arctan x for 0 < arctan x <(π/2) and f(1)=(π/4). find f(x).

$$\mathrm{the}\:\mathrm{function}\:\mathrm{f}\:\mathrm{with}\:\mathrm{variable}\:\mathrm{x} \\ $$$$\mathrm{satisfies}\:\mathrm{the}\:\mathrm{equation}\: \\ $$$$\mathrm{x}^{\mathrm{2}} \:\mathrm{f}\:'\left(\mathrm{x}\right)\:+\mathrm{2x}\:\mathrm{f}\left(\mathrm{x}\right)\:=\:\mathrm{arctan}\:\mathrm{x}\:\mathrm{for}\: \\ $$$$\mathrm{0}\:<\:\mathrm{arctan}\:\mathrm{x}\:<\frac{\pi}{\mathrm{2}}\:\mathrm{and}\:\mathrm{f}\left(\mathrm{1}\right)=\frac{\pi}{\mathrm{4}}. \\ $$$$\mathrm{find}\:\mathrm{f}\left(\mathrm{x}\right). \\ $$

Question Number 140890 Answers: 1 Comments: 0

1+2x+3x^2 +4x^3 +...+(n+1)x^n =?

$$\mathrm{1}+\mathrm{2}{x}+\mathrm{3}{x}^{\mathrm{2}} +\mathrm{4}{x}^{\mathrm{3}} +...+\left({n}+\mathrm{1}\right){x}^{\boldsymbol{{n}}} =? \\ $$

Question Number 140874 Answers: 2 Comments: 1

Question Number 140869 Answers: 1 Comments: 0

the velocities of air particles above and below the wing of an aircraft speeding down the runway at a given instant are 210m/s and 200m/s respectively.If the density of air is 1.2kg/m^3 ,what is the pressure difference between the upper and lower surface of the wing?

$${the}\:\mathrm{velocities}\:\mathrm{of}\:\mathrm{air}\:\mathrm{particles}\: \\ $$$$\mathrm{above}\:\mathrm{and}\:\mathrm{below}\:\mathrm{the}\:\mathrm{wing}\:\mathrm{of}\:\mathrm{an}\:\mathrm{aircraft} \\ $$$$\:\mathrm{speeding}\:\mathrm{down}\:\mathrm{the}\:\mathrm{runway}\:\mathrm{at}\:\mathrm{a} \\ $$$$\:\mathrm{given}\:\mathrm{instant} \\ $$$$\mathrm{are}\:\mathrm{210m}/\mathrm{s}\:\mathrm{and}\:\mathrm{200m}/\mathrm{s}\: \\ $$$$\mathrm{respectively}.\mathrm{If}\:\mathrm{the}\:\mathrm{density}\:\mathrm{of}\:\mathrm{air}\:\mathrm{is}\: \\ $$$$\mathrm{1}.\mathrm{2kg}/\mathrm{m}^{\mathrm{3}} ,\mathrm{what}\:\mathrm{is}\:\mathrm{the}\:\mathrm{pressure} \\ $$$$\:\mathrm{difference}\:\mathrm{between}\:\mathrm{the}\:\mathrm{upper}\:\mathrm{and} \\ $$$$\mathrm{lower}\:\mathrm{surface}\:\mathrm{of}\:\mathrm{the}\:\mathrm{wing}? \\ $$$$ \\ $$$$ \\ $$$$ \\ $$

Question Number 140866 Answers: 1 Comments: 0

∫_0 ^1 ((ln 2−ln (1+x^2 ))/(1−x)) dx =?

$$\underset{\mathrm{0}} {\overset{\mathrm{1}} {\int}}\:\frac{\mathrm{ln}\:\mathrm{2}−\mathrm{ln}\:\left(\mathrm{1}+\mathrm{x}^{\mathrm{2}} \right)}{\mathrm{1}−\mathrm{x}}\:\mathrm{dx}\:=?\: \\ $$

Question Number 140870 Answers: 1 Comments: 0

Ocean waves are observed to travel along the water surface during a developing storm. A Coast Guard weather station observes that there is a vertical distance from high point to low point of 4.6 meters and horizontal distance of 8.6 meters between adjacent crests.The waves splash into the station once every 6.2 seconds Determine the frequency and the speeed of these waves.

$${O}\mathrm{cean}\:\mathrm{waves}\:\mathrm{are}\:\mathrm{observed}\:\mathrm{to} \\ $$$$\mathrm{travel}\:\mathrm{along}\:\mathrm{the}\:\mathrm{water}\:\mathrm{surface} \\ $$$$\:\mathrm{during}\:\mathrm{a}\:\mathrm{developing}\:\mathrm{storm}. \\ $$$$\mathrm{A}\:\mathrm{Coast}\:\mathrm{Guard}\:\mathrm{weather}\:\mathrm{station} \\ $$$$\:\mathrm{observes}\:\mathrm{that}\:\mathrm{there}\:\mathrm{is}\:\mathrm{a}\:\mathrm{vertical}\: \\ $$$$\mathrm{distance}\:\mathrm{from}\:\mathrm{high}\:\mathrm{point}\:\mathrm{to}\:\mathrm{low} \\ $$$$\:\mathrm{point}\:\mathrm{of}\:\mathrm{4}.\mathrm{6}\:\mathrm{meters}\:\mathrm{and}\:\mathrm{horizontal} \\ $$$$\mathrm{distance}\:\mathrm{of}\:\mathrm{8}.\mathrm{6}\:\mathrm{meters}\:\mathrm{between}\: \\ $$$$\mathrm{adjacent}\:\mathrm{crests}.\mathrm{The}\:\mathrm{waves}\:\mathrm{splash}\: \\ $$$$\mathrm{into}\:\mathrm{the}\:\mathrm{station}\:\mathrm{once}\:\mathrm{every}\:\mathrm{6}.\mathrm{2}\:\mathrm{seconds} \\ $$$$\mathrm{Determine}\:\mathrm{the}\:\mathrm{frequency}\:\mathrm{and} \\ $$$$\:\mathrm{the}\:\mathrm{speeed}\:\mathrm{of}\:\mathrm{these}\:\mathrm{waves}. \\ $$

Question Number 140863 Answers: 1 Comments: 4

1+(√3^a ) = 2(√2^a )

$$\mathrm{1}+\sqrt{\mathrm{3}^{\boldsymbol{{a}}} }\:=\:\mathrm{2}\sqrt{\mathrm{2}^{\boldsymbol{{a}}} } \\ $$

Question Number 140888 Answers: 1 Comments: 0

Question Number 140885 Answers: 1 Comments: 0

....... Advanced ::::::::::★★★:::::::::: Calculus....... find the value of the infinite series:: Θ := Σ_(n=1) ^∞ (((−1)^(n−1) H_( 2n) )/(2n−1)) = ??? .......M.N.july.1970........

$$\:\:\:\:\: \\ $$$$\:\:\:\:\:\:\:\:\:\:\:.......\:{Advanced}\:::::::::::\bigstar\bigstar\bigstar::::::::::\:{Calculus}....... \\ $$$$\:\:\:\:\:\:{find}\:{the}\:{value}\:{of}\:\:{the}\:{infinite}\:{series}:: \\ $$$$\:\:\:\:\:\: \\ $$$$\:\:\:\:\:\:\:\:\:\Theta\::=\:\underset{{n}=\mathrm{1}} {\overset{\infty} {\sum}}\frac{\left(−\mathrm{1}\right)^{{n}−\mathrm{1}} \mathrm{H}_{\:\mathrm{2}{n}} }{\mathrm{2}{n}−\mathrm{1}}\:=\:??? \\ $$$$\:\:\:\:\:\:\:\:\:\:.......\mathscr{M}.\mathscr{N}.{july}.\mathrm{1970}........ \\ $$

Question Number 140838 Answers: 0 Comments: 4

Question Number 140834 Answers: 0 Comments: 4

hi, sirs ! please, how to solve this thing below (a ∈ R) { ((x_1 −x_2 = a)),((x_2 −x_3 = 2a)),((.................)),((.................)),((................)),((x_(n−1) − x_n = (n−1)a)),((x_1 + x_2 + ... + x_(n−1) + x_n = na)) :}

$$\boldsymbol{\mathrm{hi}},\:\boldsymbol{\mathrm{sirs}}\:! \\ $$$$\boldsymbol{\mathrm{please}},\:\boldsymbol{\mathrm{how}}\:\boldsymbol{\mathrm{to}}\:\boldsymbol{\mathrm{solve}}\:\boldsymbol{\mathrm{this}}\:\boldsymbol{\mathrm{thing}}\:\boldsymbol{\mathrm{below}}\:\left(\boldsymbol{{a}}\:\in\:\mathbb{R}\right)\: \\ $$$$\:\:\:\:\:\:\:\:\begin{cases}{\boldsymbol{{x}}_{\mathrm{1}} −\boldsymbol{{x}}_{\mathrm{2}} \:=\:\boldsymbol{{a}}}\\{\boldsymbol{{x}}_{\mathrm{2}} −\boldsymbol{{x}}_{\mathrm{3}} \:=\:\mathrm{2}\boldsymbol{{a}}}\\{.................}\\{.................}\\{................}\\{\boldsymbol{{x}}_{\boldsymbol{{n}}−\mathrm{1}} −\:\boldsymbol{{x}}_{\boldsymbol{{n}}} \:=\:\left(\boldsymbol{{n}}−\mathrm{1}\right)\boldsymbol{{a}}}\\{\boldsymbol{{x}}_{\mathrm{1}} +\:\boldsymbol{{x}}_{\mathrm{2}} +\:...\:+\:\boldsymbol{{x}}_{\boldsymbol{{n}}−\mathrm{1}} +\:\boldsymbol{{x}}_{\boldsymbol{{n}}} \:=\:\boldsymbol{{na}}}\end{cases} \\ $$$$ \\ $$$$ \\ $$

Question Number 140833 Answers: 3 Comments: 1

Determine if the series Σ_(n=1) ^∞ a_n defined by the formula converges or diverges . a_1 = 4 , a_(n+1) = ((10+sin n)/n). a_n

$$\mathrm{Determine}\:\mathrm{if}\:\mathrm{the}\:\mathrm{series}\:\underset{\mathrm{n}=\mathrm{1}} {\overset{\infty} {\sum}}\:\mathrm{a}_{\mathrm{n}} \\ $$$$\mathrm{defined}\:\mathrm{by}\:\mathrm{the}\:\:\mathrm{formula}\:\mathrm{converges}\:\mathrm{or} \\ $$$$\mathrm{diverges}\:.\:\mathrm{a}_{\mathrm{1}} =\:\mathrm{4}\:,\:\mathrm{a}_{\mathrm{n}+\mathrm{1}} \:=\:\frac{\mathrm{10}+\mathrm{sin}\:\mathrm{n}}{\mathrm{n}}.\:\mathrm{a}_{\mathrm{n}} \\ $$

Question Number 140831 Answers: 1 Comments: 0

......nice .... calculus...... prove that:: ξ := ∫_(−∞) ^( ∞) ((cos (πx^2 ))/(cosh(πx)))dx=(1/( (√2))) .... .......

$$\:\:\:\:\:\:\:\: \\ $$$$\:\:\:\:\:\:\:\:\:\:......{nice}\:....\:{calculus}...... \\ $$$$\:\:\:\:\:{prove}\:\:{that}:: \\ $$$$\:\:\:\:\:\:\:\:\:\xi\::=\:\int_{−\infty} ^{\:\infty} \frac{{cos}\:\left(\pi{x}^{\mathrm{2}} \right)}{{cosh}\left(\pi{x}\right)}{dx}=\frac{\mathrm{1}}{\:\sqrt{\mathrm{2}}}\:.... \\ $$$$\:\:\:\:....... \\ $$

Question Number 140829 Answers: 1 Comments: 0

factorise 5x^(2m) 8x^m +8

$${factorise}\:\mathrm{5}{x}^{\mathrm{2}{m}} \mathrm{8}{x}^{{m}} +\mathrm{8} \\ $$

Question Number 140828 Answers: 2 Comments: 0

factorise 3k^2 +2kh−8h^2

$$\mathrm{factorise}\:\mathrm{3k}^{\mathrm{2}} \:+\mathrm{2kh}−\mathrm{8h}^{\mathrm{2}} \\ $$

Question Number 140827 Answers: 1 Comments: 0

Find 1. ∫(dx/([x]^2 )) , x≥1 2. ∫(([x]^λ )/x^(λ+1) ) , x≥1 Where [∗] denote the integer part

$${Find} \\ $$$$\mathrm{1}.\:\int\frac{{dx}}{\left[{x}\right]^{\mathrm{2}} }\:,\:{x}\geqslant\mathrm{1} \\ $$$$\mathrm{2}.\:\int\frac{\left[{x}\right]^{\lambda} }{{x}^{\lambda+\mathrm{1}} }\:,\:{x}\geqslant\mathrm{1} \\ $$$${Where}\:\left[\ast\right]\:{denote}\:{the}\:{integer}\:{part} \\ $$

Question Number 140826 Answers: 0 Comments: 0

Question Number 140825 Answers: 1 Comments: 0

Use the limit comparison test to determine if the series converges or diverges Σ_(n=2) ^∞ (1/(7+8n ln (ln n))).

$$\mathrm{Use}\:\mathrm{the}\:\mathrm{limit}\:\mathrm{comparison}\:\mathrm{test} \\ $$$$\mathrm{to}\:\mathrm{determine}\:\mathrm{if}\:\mathrm{the}\:\mathrm{series}\:\mathrm{converges} \\ $$$$\mathrm{or}\:\mathrm{diverges}\: \\ $$$$\:\underset{\mathrm{n}=\mathrm{2}} {\overset{\infty} {\sum}}\:\frac{\mathrm{1}}{\mathrm{7}+\mathrm{8n}\:\mathrm{ln}\:\left(\mathrm{ln}\:\mathrm{n}\right)}.\: \\ $$

Question Number 140823 Answers: 1 Comments: 1

Determine whether the improper integral converges or diverges ∫_1 ^( ∞) ((2x+7)/(7x^3 +5x^2 +1)) dx

$$\mathrm{Determine}\:\mathrm{whether}\:\mathrm{the}\:\mathrm{improper} \\ $$$$\mathrm{integral}\:\mathrm{converges}\:\mathrm{or}\:\mathrm{diverges}\: \\ $$$$\int_{\mathrm{1}} ^{\:\infty} \:\frac{\mathrm{2x}+\mathrm{7}}{\mathrm{7x}^{\mathrm{3}} +\mathrm{5x}^{\mathrm{2}} +\mathrm{1}}\:\mathrm{dx}\: \\ $$

Question Number 140821 Answers: 0 Comments: 1

O is centre of circle and square. find yellow area in terms of radius of circle

$${O}\:{is}\:{centre}\:{of}\:{circle}\:{and}\:{square}. \\ $$$${find}\:{yellow}\:{area}\:{in}\:{terms}\:{of}\:{radius}\: \\ $$$${of}\:{circle} \\ $$

Question Number 140816 Answers: 0 Comments: 0

Question Number 140815 Answers: 0 Comments: 0

Question Number 140813 Answers: 0 Comments: 0

Question Number 142208 Answers: 1 Comments: 0

Find the equation of the circle which is orthogonal to the circles x^2 +y^2 −7x−y=0 and x^2 +y^2 +3x−6y+5=0 and which passes through the point (−3,0).

$$\mathrm{Find}\:\mathrm{the}\:\mathrm{equation}\:\mathrm{of}\:\mathrm{the}\:\mathrm{circle}\:\mathrm{which}\:\mathrm{is} \\ $$$$\mathrm{orthogonal}\:\mathrm{to}\:\mathrm{the}\:\mathrm{circles}\:{x}^{\mathrm{2}} +{y}^{\mathrm{2}} −\mathrm{7}{x}−{y}=\mathrm{0} \\ $$$$\mathrm{and}\:{x}^{\mathrm{2}} +{y}^{\mathrm{2}} +\mathrm{3}{x}−\mathrm{6}{y}+\mathrm{5}=\mathrm{0}\:\mathrm{and}\:\mathrm{which}\:\mathrm{passes} \\ $$$$\mathrm{through}\:\mathrm{the}\:\mathrm{point}\:\left(−\mathrm{3},\mathrm{0}\right). \\ $$

Question Number 142207 Answers: 2 Comments: 0

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