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Question Number 91954    Answers: 0   Comments: 4

lim_(x→0^+ ) ((ln(sin 2x))/(ln(sin 4x)))

$$\underset{{x}\rightarrow\mathrm{0}^{+} } {\mathrm{lim}}\:\frac{\mathrm{ln}\left(\mathrm{sin}\:\mathrm{2x}\right)}{\mathrm{ln}\left(\mathrm{sin}\:\mathrm{4x}\right)} \\ $$

Question Number 91948    Answers: 0   Comments: 1

Question Number 91946    Answers: 2   Comments: 3

a particle is projected from a point at a height 3h metres above a horizontal play ground. the direction of the projectile makes an angle α with the horizontal through the point of projection. show that if th greatest height reached above the point lc projection is h metres, then the horizontal distance travelled by the particle before striking the plane is 6h cotα metres. Find the vertical and horizontal component of the speed of the particle just before it hits the ground.

$$\mathrm{a}\:\mathrm{particle}\:\mathrm{is}\:\mathrm{projected}\:\mathrm{from}\:\mathrm{a}\:\mathrm{point}\:\mathrm{at}\:\mathrm{a}\:\mathrm{height}\:\mathrm{3}{h}\:\mathrm{metres}\:\mathrm{above}\:\mathrm{a}\:\mathrm{horizontal} \\ $$$$\mathrm{play}\:\mathrm{ground}.\:\mathrm{the}\:\mathrm{direction}\:\mathrm{of}\:\mathrm{the}\:\mathrm{projectile}\:\mathrm{makes}\:\mathrm{an}\:\mathrm{angle}\:\alpha\:\mathrm{with}\:\mathrm{the} \\ $$$$\mathrm{horizontal}\:\mathrm{through}\:\mathrm{the}\:\mathrm{point}\:\mathrm{of}\:\mathrm{projection}.\:\:\mathrm{show}\:\mathrm{that}\:\mathrm{if}\:\mathrm{th}\:\mathrm{greatest} \\ $$$$\mathrm{height}\:\mathrm{reached}\:\mathrm{above}\:\mathrm{the}\:\mathrm{point}\:\mathrm{lc}\:\mathrm{projection}\:\mathrm{is}\:{h}\:\mathrm{metres},\:\mathrm{then}\:\mathrm{the}\:\mathrm{horizontal} \\ $$$$\mathrm{distance}\:\mathrm{travelled}\:\mathrm{by}\:\mathrm{the}\:\mathrm{particle}\:\mathrm{before}\:\mathrm{striking}\:\mathrm{the}\:\mathrm{plane}\:\mathrm{is}\:\mathrm{6}{h}\:\mathrm{cot}\alpha\:\mathrm{metres}. \\ $$$$\mathrm{Find}\:\mathrm{the}\:\mathrm{vertical}\:\mathrm{and}\:\mathrm{horizontal}\:\mathrm{component}\:\mathrm{of}\:\mathrm{the}\:\mathrm{speed}\:\mathrm{of}\:\mathrm{the}\:\mathrm{particle}\:\mathrm{just} \\ $$$$\mathrm{before}\:\mathrm{it}\:\mathrm{hits}\:\mathrm{the}\:\mathrm{ground}. \\ $$

Question Number 91933    Answers: 1   Comments: 2

Question Number 91936    Answers: 0   Comments: 6

how to evaluate ln(i), i=(√(−1)).

$${how}\:{to}\:{evaluate}\:{ln}\left({i}\right),\:{i}=\sqrt{−\mathrm{1}}. \\ $$

Question Number 91931    Answers: 0   Comments: 3

∫_0 ^1 ln(Γ(x)) dx

$$\int_{\mathrm{0}} ^{\mathrm{1}} {ln}\left(\Gamma\left({x}\right)\right)\:{dx} \\ $$

Question Number 91930    Answers: 0   Comments: 2

lim_(x→0) cos (1/x)=

$$\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}cos}\:\frac{\mathrm{1}}{{x}}= \\ $$

Question Number 91919    Answers: 0   Comments: 1

The value of sin 12° sin 48° sin 54° is

$$\mathrm{The}\:\mathrm{value}\:\mathrm{of}\:\:\mathrm{sin}\:\mathrm{12}°\:\mathrm{sin}\:\mathrm{48}°\:\mathrm{sin}\:\mathrm{54}°\:\:\mathrm{is} \\ $$

Question Number 91914    Answers: 1   Comments: 0

hi every one here i will put my solution for old question by mr.MJS ∫((√((x−1)x(x+1)))/(3x^2 −4))dx the solution by using Appell hypergeometric function

$${hi}\:{every}\:{one}\:{here}\:{i}\:{will}\:{put}\:{my}\:{solution}\:\: \\ $$$${for}\:{old}\:{question}\:{by}\:{mr}.{MJS} \\ $$$$\int\frac{\sqrt{\left({x}−\mathrm{1}\right){x}\left({x}+\mathrm{1}\right)}}{\mathrm{3}{x}^{\mathrm{2}} −\mathrm{4}}{dx} \\ $$$$ \\ $$$${the}\:{solution}\:{by}\:{using}\:\:\: \\ $$$${Appell}\:{hypergeometric}\:{function} \\ $$

Question Number 91912    Answers: 0   Comments: 2

if ∣x∣, ∣x−1∣, ∣x+1∣ are first three terms of an AP. then what is the sum of it′s first 10 terms equal to

$$\mathrm{if}\:\mid\mathrm{x}\mid,\:\mid\mathrm{x}−\mathrm{1}\mid,\:\mid\mathrm{x}+\mathrm{1}\mid\:\mathrm{are}\:\mathrm{first} \\ $$$$\mathrm{three}\:\mathrm{terms}\:\mathrm{of}\:\mathrm{an}\:\mathrm{AP}.\:\mathrm{then}\: \\ $$$$\mathrm{what}\:\mathrm{is}\:\mathrm{the}\:\mathrm{sum}\:\mathrm{of}\:\mathrm{it}'\mathrm{s}\:\mathrm{first} \\ $$$$\mathrm{10}\:\mathrm{terms}\:\mathrm{equal}\:\mathrm{to}\: \\ $$

Question Number 91911    Answers: 1   Comments: 0

y′′′′+2y′′+y=sin x

$$\mathrm{y}''''+\mathrm{2y}''+\mathrm{y}=\mathrm{sin}\:\mathrm{x}\: \\ $$

Question Number 91910    Answers: 1   Comments: 5

∫((ln(1+sin^2 x))/(sin^2 x))dx

$$\int\frac{\mathrm{ln}\left(\mathrm{1}+\mathrm{sin}^{\mathrm{2}} \mathrm{x}\right)}{\mathrm{sin}^{\mathrm{2}} \mathrm{x}}\mathrm{dx} \\ $$

Question Number 91904    Answers: 1   Comments: 3

∫_1 ^e^π ((cos(lnx))/x)dx

$$\int_{\mathrm{1}} ^{\mathrm{e}^{\pi} } \frac{\mathrm{cos}\left(\mathrm{lnx}\right)}{\mathrm{x}}\mathrm{dx} \\ $$

Question Number 91897    Answers: 0   Comments: 1

lim_(x→0) ((sin (2x−sin x))/(1−(√(1−x^2 )))) = ?

$$\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\frac{\mathrm{sin}\:\left(\mathrm{2}{x}−\mathrm{sin}\:{x}\right)}{\mathrm{1}−\sqrt{\mathrm{1}−{x}^{\mathrm{2}} }}\:=\:? \\ $$

Question Number 91892    Answers: 0   Comments: 5

∫_0 ^π (dx/(a^2 cos^2 x+b^2 sin^2 x)) ?

$$\underset{\mathrm{0}} {\overset{\pi} {\int}}\:\frac{{dx}}{{a}^{\mathrm{2}} \mathrm{cos}\:^{\mathrm{2}} {x}+{b}^{\mathrm{2}} \:\mathrm{sin}\:^{\mathrm{2}} {x}}\:? \\ $$

Question Number 91872    Answers: 0   Comments: 3

solve in R 8(√(x^4 +1))+5(√(x^3 +1))=7x^2 +12

$${solve}\:{in}\:\mathbb{R} \\ $$$$\mathrm{8}\sqrt{\mathrm{x}^{\mathrm{4}} +\mathrm{1}}+\mathrm{5}\sqrt{\mathrm{x}^{\mathrm{3}} +\mathrm{1}}=\mathrm{7x}^{\mathrm{2}} +\mathrm{12} \\ $$

Question Number 91870    Answers: 0   Comments: 5

(1^ /x^(2x) ) + x^(−4x) = 6, (x ≠ 0) x = ?_

$$\:\frac{\overset{\:} {\mathrm{1}}}{\mathrm{x}^{\mathrm{2x}} }\:+\:\mathrm{x}^{−\mathrm{4x}} \:=\:\mathrm{6},\:\:\left(\mathrm{x}\:\neq\:\mathrm{0}\right) \\ $$$$\: \\ $$$$\:\mathrm{x}\:=\:\underset{\:} {?} \\ $$

Question Number 91862    Answers: 1   Comments: 4

log_2 (x)+log_3 (x) = 1 x =?

$$\mathrm{log}_{\mathrm{2}} \left({x}\right)+\mathrm{log}_{\mathrm{3}} \left({x}\right)\:=\:\mathrm{1} \\ $$$${x}\:=? \\ $$

Question Number 91850    Answers: 0   Comments: 3

Question Number 91848    Answers: 0   Comments: 1

y^(′′) −4y^′ +4y=(x+1)e^(2x)

$${y}^{''} −\mathrm{4}{y}^{'} +\mathrm{4}{y}=\left({x}+\mathrm{1}\right){e}^{\mathrm{2}{x}} \\ $$$$ \\ $$

Question Number 91844    Answers: 0   Comments: 5

Question Number 91843    Answers: 2   Comments: 5

{ (((1/x)+y = 2)),((x+(1/y) = 3)) :} find x^2 +y^2

$$\begin{cases}{\frac{\mathrm{1}}{{x}}+{y}\:=\:\mathrm{2}}\\{{x}+\frac{\mathrm{1}}{{y}}\:=\:\mathrm{3}}\end{cases} \\ $$$${find}\:{x}^{\mathrm{2}} +{y}^{\mathrm{2}} \\ $$

Question Number 91842    Answers: 1   Comments: 1

∫ (dx/(x(√(4x^2 +2x−1)))) ?

$$\int\:\frac{{dx}}{{x}\sqrt{\mathrm{4}{x}^{\mathrm{2}} +\mathrm{2}{x}−\mathrm{1}}}\:? \\ $$

Question Number 91840    Answers: 0   Comments: 0

solve y y′′=(y′)^2 +y′(√(y^2 +(y′)^2 ))

$${solve}\: \\ $$$${y}\:{y}''=\left({y}'\right)^{\mathrm{2}} +{y}'\sqrt{{y}^{\mathrm{2}} +\left({y}'\right)^{\mathrm{2}} } \\ $$

Question Number 91837    Answers: 0   Comments: 5

1). Σ_(n=1) ^∞ ((5/(n+2))−(5/(n+3)) )=... 2). Σ_(n=1) ^∞ ((1/(4n^2 −1)))=... 3). Σ_(n=1) ^∞ (((3n)/(5n−1)) )=... Σ_(n=1) ^∞ ((n/(n+1)) )=...

$$\left.\mathrm{1}\right).\:\:\underset{\mathrm{n}=\mathrm{1}} {\overset{\infty} {\sum}}\left(\frac{\mathrm{5}}{\mathrm{n}+\mathrm{2}}−\frac{\mathrm{5}}{\mathrm{n}+\mathrm{3}}\:\right)=... \\ $$$$ \\ $$$$\left.\mathrm{2}\right).\:\:\underset{\mathrm{n}=\mathrm{1}} {\overset{\infty} {\sum}}\left(\frac{\mathrm{1}}{\mathrm{4n}^{\mathrm{2}} −\mathrm{1}}\right)=... \\ $$$$ \\ $$$$\left.\mathrm{3}\right).\:\:\underset{\mathrm{n}=\mathrm{1}} {\overset{\infty} {\sum}}\left(\frac{\mathrm{3n}}{\mathrm{5n}−\mathrm{1}}\:\right)=... \\ $$$$ \\ $$$$\underset{\mathrm{n}=\mathrm{1}} {\overset{\infty} {\sum}}\left(\frac{\mathrm{n}}{\mathrm{n}+\mathrm{1}}\:\right)=... \\ $$

Question Number 91836    Answers: 1   Comments: 1

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