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Question Number 197343    Answers: 0   Comments: 0

calculate ∫_0 ^(π/2) ln(cosx).ln(sinx)dx

$${calculate}\:\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} {ln}\left({cosx}\right).{ln}\left({sinx}\right){dx} \\ $$

Question Number 197338    Answers: 1   Comments: 0

∫(x^2 /(x^2 +1))dx

$$\int\frac{{x}^{\mathrm{2}} }{{x}^{\mathrm{2}} +\mathrm{1}}{dx} \\ $$

Question Number 197336    Answers: 2   Comments: 0

lim_(x→+∞) ((1/x^2 )+cosx)=?

$${lim}_{{x}\rightarrow+\infty} \left(\frac{\mathrm{1}}{{x}^{\mathrm{2}} }+\mathrm{cos}{x}\right)=? \\ $$

Question Number 197327    Answers: 2   Comments: 0

trigonometry... P = Π_(k=1) ^(44) ( 1 + tan(k) ) = ?

$$ \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:{trigonometry}... \\ $$$$ \\ $$$$\:\:\:\:\:\:\:\:\:\:\mathrm{P}\:=\:\underset{{k}=\mathrm{1}} {\overset{\mathrm{44}} {\prod}}\left(\:\:\mathrm{1}\:+\:{tan}\left({k}\right)\:\right)\:=\:?\:\: \\ $$$$ \\ $$$$\:\:\:\:\:\:\:\: \\ $$

Question Number 197335    Answers: 1   Comments: 0

Question Number 197325    Answers: 1   Comments: 0

Question Number 197323    Answers: 1   Comments: 0

If f(x)=((sin(x))/x) and S_n (α)=Σ_(k=1) ^n [f(kπ+(π/α))+f(kπ−(π/α))] (α>1) Prove that lim_(n→+∞) S_n (α)=1−f((π/α))

$$\mathrm{If}\:\mathrm{f}\left(\mathrm{x}\right)=\frac{\mathrm{sin}\left(\mathrm{x}\right)}{\mathrm{x}}\:\:\:\mathrm{and}\:\mathrm{S}_{\mathrm{n}} \left(\alpha\right)=\underset{\mathrm{k}=\mathrm{1}} {\overset{\mathrm{n}} {\sum}}\left[\mathrm{f}\left(\mathrm{k}\pi+\frac{\pi}{\alpha}\right)+\mathrm{f}\left(\mathrm{k}\pi−\frac{\pi}{\alpha}\right)\right]\:\:\:\:\left(\alpha>\mathrm{1}\right) \\ $$$$\mathrm{Prove}\:\mathrm{that}\:\underset{\mathrm{n}\rightarrow+\infty} {\:\mathrm{lim}}\:\mathrm{S}_{\mathrm{n}} \left(\alpha\right)=\mathrm{1}−\mathrm{f}\left(\frac{\pi}{\alpha}\right) \\ $$

Question Number 197320    Answers: 1   Comments: 0

Σ_(k=1) ^n (−1)^(k(k+1))

$$\underset{{k}=\mathrm{1}} {\overset{{n}} {\sum}}\left(−\mathrm{1}\right)^{{k}\left({k}+\mathrm{1}\right)} \\ $$

Question Number 197317    Answers: 0   Comments: 0

if x = ((cos θ)/u) , y = ((sin θ)/u) and z = f(x,y) then show that (∂^2 z/∂x^2 ) + (∂^2 z/∂y^2 ) = u^4 (∂^2 z/∂u^2 ) + u^3 (∂z/∂u) + u^4 (∂^2 z/∂θ^2 )

$$\:\mathrm{if}\:\:\:\mathrm{x}\:\:=\:\:\frac{\mathrm{cos}\:\theta}{\mathrm{u}}\:\:,\:\mathrm{y}\:\:=\:\frac{\mathrm{sin}\:\theta}{\mathrm{u}}\:\:{and}\:\mathrm{z}\:\:=\:\:\mathrm{f}\left(\mathrm{x},\mathrm{y}\right) \\ $$$$\mathrm{then}\:\mathrm{show}\:\mathrm{that}\: \\ $$$$\:\:\:\frac{\partial^{\mathrm{2}} \mathrm{z}}{\partial\mathrm{x}^{\mathrm{2}} }\:+\:\frac{\partial^{\mathrm{2}} \mathrm{z}}{\partial\mathrm{y}^{\mathrm{2}} }\:=\:\mathrm{u}^{\mathrm{4}} \:\frac{\partial^{\mathrm{2}} \mathrm{z}}{\partial\mathrm{u}^{\mathrm{2}} }\:+\:\mathrm{u}^{\mathrm{3}} \:\frac{\partial\mathrm{z}}{\partial\mathrm{u}}\:+\:\mathrm{u}^{\mathrm{4}} \:\frac{\partial^{\mathrm{2}} \mathrm{z}}{\partial\theta^{\mathrm{2}} } \\ $$

Question Number 197312    Answers: 1   Comments: 0

(((log_2 20)^2 −(log_2 5)^2 )/(log_2 10))=?

$$\frac{\left({log}_{\mathrm{2}} \mathrm{20}\right)^{\mathrm{2}} −\left({log}_{\mathrm{2}} \mathrm{5}\right)^{\mathrm{2}} }{{log}_{\mathrm{2}} \mathrm{10}}=? \\ $$

Question Number 197311    Answers: 1   Comments: 0

Prove that _(n+1) C_r = _n C_r + _n C_(r−1)

$$\:\:\:\:\:\mathrm{Prove}\:\mathrm{that}\: \\ $$$$\:\:\:\:\:\:_{\mathrm{n}+\mathrm{1}} \:\mathrm{C}_{\mathrm{r}} \:=\:_{\mathrm{n}} \mathrm{C}_{\mathrm{r}} \:+\:_{\mathrm{n}} \mathrm{C}_{\mathrm{r}−\mathrm{1}} \: \\ $$

Question Number 197310    Answers: 1   Comments: 0

Question Number 197301    Answers: 1   Comments: 0

how do i calculate this lim_(x→-∞) ((x^4 +2x^2 +x−2)/(x^3 +2x^2 +x−1)) multiplying both numerator and denumerator by (1/x^4 ) lim_(x→-∞) ((1+(2/x^2 )+(1/x^3 )−(2/x^4 ))/((1/x)+(2/x^2 )+(1/x^3 )−(1/x^4 ))) ((1+0+0−0)/(0+0+0−0)) ∞ which is not true the answer is -∞, i tried multiplying (1/x^3 ) and got -∞ but still confused what did i do wrong using (1/x^4 )

$$ \\ $$$$\:{how}\:{do}\:{i}\:{calculate}\:{this} \\ $$$$\:\underset{{x}\rightarrow-\infty} {\mathrm{lim}}\:\frac{{x}^{\mathrm{4}} +\mathrm{2}{x}^{\mathrm{2}} +{x}−\mathrm{2}}{{x}^{\mathrm{3}} +\mathrm{2}{x}^{\mathrm{2}} +{x}−\mathrm{1}} \\ $$$$\:{multiplying}\:{both}\:{numerator} \\ $$$$\:{and}\:{denumerator}\:{by}\:\frac{\mathrm{1}}{{x}^{\mathrm{4}} } \\ $$$$\:\underset{{x}\rightarrow-\infty} {\mathrm{lim}}\:\frac{\mathrm{1}+\frac{\mathrm{2}}{{x}^{\mathrm{2}} }+\frac{\mathrm{1}}{{x}^{\mathrm{3}} }−\frac{\mathrm{2}}{{x}^{\mathrm{4}} }}{\frac{\mathrm{1}}{{x}}+\frac{\mathrm{2}}{{x}^{\mathrm{2}} }+\frac{\mathrm{1}}{{x}^{\mathrm{3}} }−\frac{\mathrm{1}}{{x}^{\mathrm{4}} }} \\ $$$$\:\frac{\mathrm{1}+\mathrm{0}+\mathrm{0}−\mathrm{0}}{\mathrm{0}+\mathrm{0}+\mathrm{0}−\mathrm{0}} \\ $$$$\:\infty \\ $$$$\:{which}\:{is}\:{not}\:{true}\:{the}\:{answer}\:{is}\:-\infty, \\ $$$$\:{i}\:{tried}\:{multiplying}\:\frac{\mathrm{1}}{{x}^{\mathrm{3}} }\:{and}\:{got}\:-\infty \\ $$$$\:{but}\:{still}\:{confused}\:{what}\:{did}\:{i}\:{do}\:{wrong} \\ $$$$\:{using}\:\frac{\mathrm{1}}{{x}^{\mathrm{4}} } \\ $$$$ \\ $$

Question Number 197299    Answers: 1   Comments: 0

((−64))^(1/6) −((−10))^(1/(10)) =? I need so much plz

$$\sqrt[{\mathrm{6}}]{−\mathrm{64}}−\sqrt[{\mathrm{10}}]{−\mathrm{10}}=? \\ $$$$\boldsymbol{{I}}\:\boldsymbol{\mathrm{need}}\:\boldsymbol{\mathrm{so}}\:\boldsymbol{\mathrm{much}}\:\boldsymbol{\mathrm{plz}} \\ $$

Question Number 197292    Answers: 2   Comments: 0

lim_(n→∞) ∫_(0 ) ^1 ((nx^(n−1) )/(1+x))dx = ?

$$\:\:\:\underset{{n}\rightarrow\infty} {\mathrm{lim}}\:\int_{\mathrm{0}\:} ^{\mathrm{1}} \frac{{nx}^{{n}−\mathrm{1}} }{\mathrm{1}+{x}}{dx}\:\:=\:\:\:? \\ $$

Question Number 197290    Answers: 0   Comments: 1

Question Number 197287    Answers: 0   Comments: 3

answer to the question number 197017 AF=FI & AG=GJ⇒FG=(1/2)IJ=(1/6)BC △FGH is squilatral ⇒ △FGH≈△ABC ⇒(S_(FGH) /S_(SBC) ) =(1/(36 )) ✓

$${answer}\:{to}\:{the}\:{question}\:{number} \\ $$$$\mathrm{197017} \\ $$$${AF}={FI}\:\&\:\:{AG}={GJ}\Rightarrow{FG}=\frac{\mathrm{1}}{\mathrm{2}}{IJ}=\frac{\mathrm{1}}{\mathrm{6}}{BC} \\ $$$$\bigtriangleup{FGH}\:\:{is}\:\:{squilatral}\:\Rightarrow\:\bigtriangleup{FGH}\approx\bigtriangleup{ABC} \\ $$$$\Rightarrow\frac{{S}_{{FGH}} }{{S}_{{SBC}} }\:=\frac{\mathrm{1}}{\mathrm{36}\:}\:\checkmark \\ $$$$ \\ $$

Question Number 197282    Answers: 1   Comments: 0

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

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

Question Number 197281    Answers: 2   Comments: 0

lim_(x→0) ((sin x−x+2x^5 )/(3x^3 )) =?

$$\:\:\:\:\:\:\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\frac{\mathrm{sin}\:\mathrm{x}−\mathrm{x}+\mathrm{2x}^{\mathrm{5}} }{\mathrm{3x}^{\mathrm{3}} }\:=? \\ $$

Question Number 197277    Answers: 1   Comments: 0

Question Number 197275    Answers: 1   Comments: 0

how do i prove this, help please. ∣((x^2 −2x−3)/(x^2 +2x+4))∣≤(5/4),∣x∣≤2

$$ \\ $$$$\:{how}\:{do}\:{i}\:{prove}\:{this},\:{help}\:{please}. \\ $$$$\:\mid\frac{{x}^{\mathrm{2}} −\mathrm{2}{x}−\mathrm{3}}{{x}^{\mathrm{2}} +\mathrm{2}{x}+\mathrm{4}}\mid\leqslant\frac{\mathrm{5}}{\mathrm{4}},\mid{x}\mid\leqslant\mathrm{2} \\ $$$$ \\ $$$$ \\ $$

Question Number 197274    Answers: 0   Comments: 1

Question Number 197272    Answers: 2   Comments: 0

How to calculate this integral ∫^( (π/2)) _( 0) ((ln(1+sint))/(sint))dt

$$\mathrm{How}\:\mathrm{to}\:\mathrm{calculate}\:\mathrm{this}\:\mathrm{integral} \\ $$$$\underset{\:\mathrm{0}} {\int}^{\:\frac{\pi}{\mathrm{2}}} \:\frac{\mathrm{ln}\left(\mathrm{1}+{sint}\right)}{{sint}}{dt} \\ $$

Question Number 198332    Answers: 0   Comments: 1

Of men that attended a party, 30 of them wore coats, 20 wore ties and 10 wore hats. There were 4 men who wore coats and tie, or tie and hat or coat and hat. 14 men wore tie only with no coat and hat. Find the number of men who wore: a) coat, tie and hat b) hat only with no coat and hat.

$${Of}\:{men}\:{that}\:{attended}\:{a}\:{party},\:\mathrm{30}\:{of} \\ $$$${them}\:{wore}\:{coats},\:\mathrm{20}\:{wore}\:{ties}\:{and}\:\mathrm{10} \\ $$$${wore}\:{hats}.\:{There}\:{were}\:\mathrm{4}\:{men}\:{who}\:{wore} \\ $$$${coats}\:{and}\:{tie},\:{or}\:{tie}\:{and}\:{hat}\:{or}\:{coat}\:{and} \\ $$$${hat}.\:\mathrm{14}\:{men}\:{wore}\:{tie}\:{only}\:{with}\:{no} \\ $$$${coat}\:{and}\:{hat}.\:{Find}\:{the}\:{number}\:{of}\:{men} \\ $$$${who}\:{wore}: \\ $$$$\left.{a}\right)\:{coat},\:{tie}\:{and}\:{hat} \\ $$$$\left.{b}\right)\:{hat}\:{only}\:{with}\:{no}\:{coat}\:{and}\:{hat}. \\ $$

Question Number 198288    Answers: 1   Comments: 0

Question Number 197255    Answers: 0   Comments: 0

log 2=a log 3=b log 72=?

$$\mathrm{log}\:\mathrm{2}={a} \\ $$$$\mathrm{log}\:\mathrm{3}={b} \\ $$$$\mathrm{log}\:\mathrm{72}=? \\ $$

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