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Question Number 114145 Answers: 1 Comments: 0

A particle moves along a straight line such that its velocity, v m s^(−1) , is given by v=t^3 −4t^2 +3t, where t is time, in seconds, after passing through fixed point O. Find the total distance, in m, travelled by the particle until the particle returned to the fixed point O for the second time.

$$\mathrm{A}\:\mathrm{particle}\:\mathrm{moves}\:\mathrm{along}\:\mathrm{a}\:\mathrm{straight}\:\mathrm{line}\: \\ $$$$\mathrm{such}\:\mathrm{that}\:\mathrm{its}\:\mathrm{velocity},\:{v}\:\mathrm{m}\:\mathrm{s}^{−\mathrm{1}} ,\:\mathrm{is}\:\mathrm{given} \\ $$$$\mathrm{by}\:{v}={t}^{\mathrm{3}} −\mathrm{4}{t}^{\mathrm{2}} +\mathrm{3}{t},\:\mathrm{where}\:{t}\:\mathrm{is}\:\mathrm{time},\:\mathrm{in}\: \\ $$$$\mathrm{seconds},\:\mathrm{after}\:\mathrm{passing}\:\mathrm{through}\:\mathrm{fixed} \\ $$$$\mathrm{point}\:\mathrm{O}. \\ $$$$\mathrm{Find}\:\mathrm{the}\:\mathrm{total}\:\mathrm{distance},\:\mathrm{in}\:\mathrm{m},\:\mathrm{travelled} \\ $$$$\mathrm{by}\:\mathrm{the}\:\mathrm{particle}\:\mathrm{until}\:\mathrm{the}\:\mathrm{particle}\:\mathrm{returned} \\ $$$$\mathrm{to}\:\mathrm{the}\:\mathrm{fixed}\:\mathrm{point}\:\mathrm{O}\:\mathrm{for}\:\mathrm{the}\:\mathrm{second}\:\mathrm{time}. \\ $$

Question Number 114135 Answers: 2 Comments: 0

....Advanced mathematics ... i:: prove that : Ω=(1/π)∫_0 ^( ∞) (1/((x^2 −x+1)^2 (√x)))dx =1 ii::evaluate :: Φ = ∫_0 ^( 1) x^2 ln(x) ln(1−x)dx=??? ....m.n.july. 1970....

$$\:\:\:\:\:\:\:\:\:\:\:\:....\mathscr{A}{dvanced}\:\:{mathematics}\:... \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:{i}::\:{prove}\:\:{that}\::\:\:\:\:\Omega=\frac{\mathrm{1}}{\pi}\int_{\mathrm{0}} ^{\:\infty} \frac{\mathrm{1}}{\left({x}^{\mathrm{2}} −{x}+\mathrm{1}\right)^{\mathrm{2}} \sqrt{{x}}}{dx}\:=\mathrm{1}\:\: \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:{ii}::{evaluate}\:::\:\:\Phi\:=\:\int_{\mathrm{0}} ^{\:\mathrm{1}} {x}^{\mathrm{2}} \:{ln}\left({x}\right)\:{ln}\left(\mathrm{1}−{x}\right){dx}=??? \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:....{m}.{n}.{july}.\:\mathrm{1970}.... \\ $$$$\: \\ $$

Question Number 114134 Answers: 1 Comments: 4

(m^2 −n^2 +6(n+m)/(m^2 −(6−n)^2 m+n=12

$$\left({m}^{\mathrm{2}} −{n}^{\mathrm{2}} +\mathrm{6}\left({n}+{m}\right)/\left({m}^{\mathrm{2}} −\left(\mathrm{6}−{n}\right)^{\mathrm{2}} \right.\right. \\ $$$${m}+{n}=\mathrm{12} \\ $$

Question Number 114130 Answers: 1 Comments: 5

Question Number 114122 Answers: 1 Comments: 0

find the value sin (cos^(−1) ((3/5))+tan^(−1) ((7/(13))))

$${find}\:{the}\:{value}\:\mathrm{sin}\:\left(\mathrm{cos}^{−\mathrm{1}} \left(\frac{\mathrm{3}}{\mathrm{5}}\right)+\mathrm{tan}^{−\mathrm{1}} \left(\frac{\mathrm{7}}{\mathrm{13}}\right)\right) \\ $$

Question Number 114114 Answers: 1 Comments: 1

A 50g golf ball is struck with a club moving with a velocity of 22m/s after it moves 4cm and ball accelerates with a velocity of 44m/s. estimate the average force exerted by the club on the ball.

$$ \\ $$$$\mathrm{A}\:\mathrm{50g}\:\mathrm{golf}\:\mathrm{ball}\:\mathrm{is}\:\mathrm{struck}\:\mathrm{with}\:\mathrm{a}\:\mathrm{club}\:\mathrm{moving} \\ $$$$\mathrm{with}\:\mathrm{a}\:\mathrm{velocity}\:\mathrm{of}\:\mathrm{22m}/\mathrm{s}\:\mathrm{after}\:\mathrm{it}\:\mathrm{moves}\:\mathrm{4cm}\:\mathrm{and} \\ $$$$\mathrm{ball}\:\mathrm{accelerates}\:\mathrm{with}\:\mathrm{a}\:\mathrm{velocity}\:\mathrm{of}\:\mathrm{44m}/\mathrm{s}. \\ $$$$\mathrm{estimate}\:\mathrm{the}\:\mathrm{average}\:\mathrm{force}\:\mathrm{exerted}\:\mathrm{by}\:\mathrm{the}\:\mathrm{club} \\ $$$$\mathrm{on}\:\mathrm{the}\:\mathrm{ball}. \\ $$

Question Number 114110 Answers: 1 Comments: 2

(4/(1!)) + ((11)/(2!)) + ((22)/(3!)) + ((37)/(4!)) + ... = ?

$$\frac{\mathrm{4}}{\mathrm{1}!}\:+\:\frac{\mathrm{11}}{\mathrm{2}!}\:+\:\frac{\mathrm{22}}{\mathrm{3}!}\:+\:\frac{\mathrm{37}}{\mathrm{4}!}\:+\:...\:=\:? \\ $$

Question Number 114108 Answers: 3 Comments: 0

prove that 2 tan^(−1) ((2/3))=sin^(−1) (((12)/(13)))

$${prove}\:{that}\:\mathrm{2}\:\mathrm{tan}^{−\mathrm{1}} \left(\frac{\mathrm{2}}{\mathrm{3}}\right)=\mathrm{sin}^{−\mathrm{1}} \left(\frac{\mathrm{12}}{\mathrm{13}}\right) \\ $$

Question Number 114107 Answers: 1 Comments: 2

prove that ∫_0 ^(π/2) [((ln(((1−sinx)/(1+sinx)))(√(cosx)))/((1+sinx)(√(1−sinx))))]dx=−8

$${prove}\:{that} \\ $$$$\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \left[\frac{\mathrm{ln}\left(\frac{\mathrm{1}−\mathrm{sin}{x}}{\mathrm{1}+\mathrm{sin}{x}}\right)\sqrt{\mathrm{cos}{x}}}{\left(\mathrm{1}+\mathrm{sin}{x}\right)\sqrt{\mathrm{1}−\mathrm{sin}{x}}}\right]{dx}=−\mathrm{8} \\ $$

Question Number 114105 Answers: 0 Comments: 1

∫ ((arctan(e^x ))/( (√x))) dx

$$\int\:\frac{{arctan}\left({e}^{{x}} \right)}{\:\sqrt{{x}}}\:{dx} \\ $$

Question Number 114103 Answers: 0 Comments: 1

∫(√(ln(tan(x))))dx

$$\int\sqrt{{ln}\left({tan}\left({x}\right)\right)}{dx} \\ $$

Question Number 114102 Answers: 2 Comments: 0

∫ (dx/(tan x−sin x))

$$\int\:\frac{{dx}}{\mathrm{tan}\:{x}−\mathrm{sin}\:{x}} \\ $$

Question Number 114101 Answers: 0 Comments: 0

Let x are positive even numbers more than 8 . What is the remainder of divisibility : (x + 1)^x − x^x by (x + 2)^2

$${Let}\:\:{x}\:\:{are}\:\:{positive}\:\:{even}\:\:{numbers}\:\:{more}\:\:{than}\:\:\mathrm{8}\:. \\ $$$${What}\:\:{is}\:\:{the}\:\:{remainder}\:\:{of}\:\:\:{divisibility}\:\:: \\ $$$$\left({x}\:+\:\mathrm{1}\right)^{{x}} \:−\:{x}^{{x}} \:\:{by}\:\:\left({x}\:+\:\mathrm{2}\right)^{\mathrm{2}} \\ $$

Question Number 114100 Answers: 1 Comments: 2

If cos =((acos α−b)/(a−bcos α)), prove that, ((tan (1/2)θ)/( (√(a+b))))=((tan (1/2)α)/( (√(a+b))))

$$ \\ $$$${If}\:\mathrm{cos}\:=\frac{{a}\mathrm{cos}\:\alpha−{b}}{{a}−{b}\mathrm{cos}\:\alpha},\:{prove}\:{that}, \\ $$$$\frac{\mathrm{tan}\:\frac{\mathrm{1}}{\mathrm{2}}\theta}{\:\sqrt{{a}+{b}}}=\frac{\mathrm{tan}\:\frac{\mathrm{1}}{\mathrm{2}}\alpha}{\:\sqrt{{a}+{b}}} \\ $$

Question Number 114099 Answers: 1 Comments: 0

.... mathematical analysis.... prove that :: Σ_(n=1) ^∞ ( ((3^n −1)/4^n ))ζ(n+1) =π m.n.july.1970#

$$\:\:\:....\:\:{mathematical}\:\:{analysis}....\:\: \\ $$$$\:\:{prove}\:{that}\::: \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\: \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\underset{{n}=\mathrm{1}} {\overset{\infty} {\sum}}\left(\:\frac{\mathrm{3}^{{n}} −\mathrm{1}}{\mathrm{4}^{{n}} }\right)\zeta\left({n}+\mathrm{1}\right)\:=\pi\: \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:{m}.{n}.{july}.\mathrm{1970}# \\ $$$$ \\ $$

Question Number 114094 Answers: 3 Comments: 0

∫ln(x)sin^(−1) (x)dx

$$\int\boldsymbol{\mathrm{ln}}\left(\boldsymbol{\mathrm{x}}\right)\boldsymbol{\mathrm{sin}}^{−\mathrm{1}} \left(\boldsymbol{\mathrm{x}}\right)\boldsymbol{\mathrm{dx}} \\ $$

Question Number 114088 Answers: 1 Comments: 0

lim_(x→∞) ((2x cot ((2/x))−3cot ((2/x)))/(5x^2 −2x))

$$\underset{{x}\rightarrow\infty} {\mathrm{lim}}\:\frac{\mathrm{2}{x}\:\mathrm{cot}\:\left(\frac{\mathrm{2}}{{x}}\right)−\mathrm{3cot}\:\left(\frac{\mathrm{2}}{{x}}\right)}{\mathrm{5}{x}^{\mathrm{2}} −\mathrm{2}{x}} \\ $$

Question Number 114079 Answers: 2 Comments: 0

Question Number 114077 Answers: 1 Comments: 0

Question Number 114072 Answers: 3 Comments: 0

∫(1/(sinx + cosx))dx

$$\int\frac{\mathrm{1}}{\boldsymbol{\mathrm{sinx}}\:+\:\boldsymbol{\mathrm{cosx}}}\boldsymbol{\mathrm{dx}} \\ $$

Question Number 114071 Answers: 1 Comments: 4

Find the solution set (√(x^2 −4x−5)) ≥ x

$${Find}\:{the}\:{solution}\:{set}\: \\ $$$$\sqrt{{x}^{\mathrm{2}} −\mathrm{4}{x}−\mathrm{5}}\:\geqslant\:{x} \\ $$

Question Number 114066 Answers: 2 Comments: 1

Given f(x)=7+cos 2x+2sin^2 x find f^((10)) (x) ?

$${Given}\:{f}\left({x}\right)=\mathrm{7}+\mathrm{cos}\:\mathrm{2}{x}+\mathrm{2sin}\:^{\mathrm{2}} {x} \\ $$$${find}\:{f}^{\left(\mathrm{10}\right)} \left({x}\right)\:? \\ $$

Question Number 114061 Answers: 1 Comments: 0

Question Number 114075 Answers: 2 Comments: 0

Question Number 114056 Answers: 4 Comments: 0

calculate ∫_2 ^(+∞) (dt/((2t+3)^4 (t−1)^5 ))

$$\mathrm{calculate}\:\int_{\mathrm{2}} ^{+\infty} \:\:\:\:\frac{\mathrm{dt}}{\left(\mathrm{2t}+\mathrm{3}\right)^{\mathrm{4}} \left(\mathrm{t}−\mathrm{1}\right)^{\mathrm{5}} } \\ $$

Question Number 114045 Answers: 4 Comments: 0

... advanced calculus... i : prove that :: ∫_0 ^( 1) ((ln(1+ln(1−x)))/(ln(1−x))) dx =^? Σ_(n=1) ^∞ ((Γ(n+1))/n^2 ) ii: prove that :: Ω =∫_0 ^( 1) ((ln(1+x))/(x(1+x^2 )))dx =^? ((5π^2 )/(48)) m.n.july 1970#

$$\:\:\:\:\:\:\:\:...\:\:{advanced}\:{calculus}... \\ $$$$ \\ $$$${i}\::\:\:{prove}\:\:{that}\::: \\ $$$$\:\int_{\mathrm{0}} ^{\:\mathrm{1}} \frac{{ln}\left(\mathrm{1}+{ln}\left(\mathrm{1}−{x}\right)\right)}{{ln}\left(\mathrm{1}−{x}\right)}\:{dx}\:\overset{?} {=}\underset{{n}=\mathrm{1}} {\overset{\infty} {\sum}}\frac{\Gamma\left({n}+\mathrm{1}\right)}{{n}^{\mathrm{2}} }\:\:\:\:\:\:\:\:\:\:\:\:\:\: \\ $$$${ii}:\: \\ $$$$\:\:\:\:{prove}\:{that}\::: \\ $$$$\:\:\:\:\:\:\:\Omega\:=\int_{\mathrm{0}} ^{\:\mathrm{1}} \frac{{ln}\left(\mathrm{1}+{x}\right)}{{x}\left(\mathrm{1}+{x}^{\mathrm{2}} \right)}{dx}\:\overset{?} {=}\frac{\mathrm{5}\pi^{\mathrm{2}} }{\mathrm{48}} \\ $$$$ \\ $$$$\:\:\:\:\:\:\:\:\:\:\:{m}.{n}.{july}\:\mathrm{1970}# \\ $$$$\:\:\: \\ $$

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