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Question Number 79735    Answers: 1   Comments: 0

write tanhx in terms of e, hence prove that tanh2x = ((2tanhx)/(1+tanh^2 x))

$${write}\:{tanhx}\:{in}\:{terms}\:{of}\:{e},\:{hence}\:{prove}\:{that}\: \\ $$$${tanh}\mathrm{2}{x}\:=\:\frac{\mathrm{2}{tanhx}}{\mathrm{1}+{tanh}^{\mathrm{2}} {x}} \\ $$

Question Number 79731    Answers: 1   Comments: 5

Question Number 79730    Answers: 1   Comments: 1

I) For witch value of α the integral C=∫_0 ^( ∞) ((1/(√(1+2x^2 )))−(1/(x+1)))dx conveege ? And in this case calculate α. II) Let Δ={(x; y)/ ∣x∣+∣y∣≤2} a) Calculate I_1 = ∫∫_Δ dxdy and ∫∫_Δ ((dxdy)/((∣x∣+∣y∣)^2 +4))

$$\left.{I}\right)\:\:{For}\:{witch}\:{value}\:{of}\:\alpha\:{the}\:{integral} \\ $$$$\:{C}=\int_{\mathrm{0}} ^{\:\infty} \left(\frac{\mathrm{1}}{\sqrt{\mathrm{1}+\mathrm{2}{x}^{\mathrm{2}} }}−\frac{\mathrm{1}}{{x}+\mathrm{1}}\right){dx}\:\:{conveege}\:\:? \\ $$$${And}\:{in}\:{this}\:{case}\:{calculate}\:\alpha. \\ $$$$\left.{II}\right)\:\:{Let}\:\Delta=\left\{\left({x};\:{y}\right)/\:\mid{x}\mid+\mid{y}\mid\leqslant\mathrm{2}\right\} \\ $$$$\left.\:\:\:\:\:{a}\right)\:{Calculate}\:{I}_{\mathrm{1}} =\:\int\int_{\Delta} {dxdy}\:\:\:{and}\:\:\int\int_{\Delta} \frac{{dxdy}}{\left(\mid{x}\mid+\mid{y}\mid\right)^{\mathrm{2}} +\mathrm{4}} \\ $$

Question Number 79756    Answers: 0   Comments: 2

calculate lim_(x→1) ((sin(πx))/(1−x^2 )) without hospital rule

$${calculate}\:{lim}_{{x}\rightarrow\mathrm{1}} \:\:\frac{{sin}\left(\pi{x}\right)}{\mathrm{1}−{x}^{\mathrm{2}} }\:\:{without}\:{hospital}\:{rule} \\ $$

Question Number 79719    Answers: 1   Comments: 1

(((2.39)^2 −(1.61)^2 )/(2.39−1.61))...?

$$\frac{\left(\mathrm{2}.\mathrm{39}\overset{\mathrm{2}} {\right)}−\left(\mathrm{1}.\mathrm{61}\overset{\mathrm{2}} {\right)}}{\mathrm{2}.\mathrm{39}−\mathrm{1}.\mathrm{61}}...? \\ $$$$ \\ $$

Question Number 79718    Answers: 0   Comments: 0

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Question Number 79705    Answers: 0   Comments: 3

if L{f(t)}=L{g(t)} then why f(t)=g(t)? is there any proof

$$\mathrm{if}\:\mathscr{L}\left\{\mathrm{f}\left(\mathrm{t}\right)\right\}=\mathscr{L}\left\{\mathrm{g}\left(\mathrm{t}\right)\right\} \\ $$$$\mathrm{then}\:\mathrm{why}\:\mathrm{f}\left(\mathrm{t}\right)=\mathrm{g}\left(\mathrm{t}\right)? \\ $$$$\mathrm{is}\:\mathrm{there}\:\mathrm{any}\:\mathrm{proof} \\ $$

Question Number 79697    Answers: 2   Comments: 1

Question Number 79694    Answers: 0   Comments: 1

Question Number 79689    Answers: 0   Comments: 2

Find lim_(x→∞) x^2 ln(1+(1/x)) −x without DL and if you have to use the hospital rule please justify that your function is C^1

$${Find}\:\:\underset{{x}\rightarrow\infty} {\mathrm{lim}}\:{x}^{\mathrm{2}} {ln}\left(\mathrm{1}+\frac{\mathrm{1}}{{x}}\right)\:−{x}\:\:\:\:\:{without}\:\:{DL}\:{and}\:{if}\:{you}\:{have}\:{to}\:{use}\:{the}\: \\ $$$${hospital}\:{rule}\:{please}\:\:{justify}\:{that}\:{your}\:{function}\:{is}\:{C}^{\mathrm{1}} \: \\ $$

Question Number 79679    Answers: 0   Comments: 2

Find lim_(x→∞) x^2 ln(1+x)−x

$${Find}\:\underset{{x}\rightarrow\infty} {\mathrm{lim}}\:{x}^{\mathrm{2}} {ln}\left(\mathrm{1}+{x}\right)−{x} \\ $$

Question Number 79675    Answers: 0   Comments: 11

Question Number 79670    Answers: 0   Comments: 0

find the covulation coefficient from the regression equation x=15.03−0.98y and y=14.72−0.93x

$${find}\:{the}\:{covulation}\:{coefficient}\:{from}\: \\ $$$${the}\:{regression}\:{equation}\:{x}=\mathrm{15}.\mathrm{03}−\mathrm{0}.\mathrm{98}{y} \\ $$$${and}\:{y}=\mathrm{14}.\mathrm{72}−\mathrm{0}.\mathrm{93}{x} \\ $$

Question Number 79652    Answers: 0   Comments: 0

what is the value of Σ_(n=1) ^∞ [(4n^2 −1)ln(1−(1/(4n^2 )))+1]

$$\mathrm{what}\:\mathrm{is}\:\mathrm{the}\:\mathrm{value}\:\mathrm{of} \\ $$$$\underset{\mathrm{n}=\mathrm{1}} {\overset{\infty} {\sum}}\:\left[\left(\mathrm{4n}^{\mathrm{2}} −\mathrm{1}\right)\mathrm{ln}\left(\mathrm{1}−\frac{\mathrm{1}}{\mathrm{4n}^{\mathrm{2}} }\right)+\mathrm{1}\right]\: \\ $$

Question Number 79649    Answers: 1   Comments: 6

given a,ar,ar^2 ,ar^3 ,... is a GPwith n→∞ ,r < 1 if : a,x_1 ,x_2 ,ar,x_3 , x_4 ,x_5 ,x_6 ,ar^2 , x_7 ,x_8 ,x_9 ,x_(10) ,x_(11) ,x_(12) , ar^3 ,... . where : a,x_1 ,x_2 ,ar ⇒AP ar,x_3 ,x_4 ,x_5 ,x_6 ,ar^2 ⇒AP ar^2 ,x_7 ,x_8 ,x_9 ,x_(10) ,x_(11) ,x_(12) ,ar^3 ⇒AP ...etc if lim_(n→∞) (x_1 +x_2 +x_3 +...)= ((21)/(16))×(a/(1−r)) what is r ?

$$\mathrm{given}\:\mathrm{a},\mathrm{ar},\mathrm{ar}^{\mathrm{2}} ,\mathrm{ar}^{\mathrm{3}} ,...\:\mathrm{is}\:\mathrm{a}\:\mathrm{GPwith}\: \\ $$$$\mathrm{n}\rightarrow\infty\:,\mathrm{r}\:<\:\mathrm{1} \\ $$$$\mathrm{if}\::\:\mathrm{a},\mathrm{x}_{\mathrm{1}} ,\mathrm{x}_{\mathrm{2}} ,\mathrm{ar},\mathrm{x}_{\mathrm{3}} ,\:\mathrm{x}_{\mathrm{4}} ,\mathrm{x}_{\mathrm{5}} ,\mathrm{x}_{\mathrm{6}} ,\mathrm{ar}^{\mathrm{2}} , \\ $$$$\mathrm{x}_{\mathrm{7}} ,\mathrm{x}_{\mathrm{8}} ,\mathrm{x}_{\mathrm{9}} ,\mathrm{x}_{\mathrm{10}} ,\mathrm{x}_{\mathrm{11}} ,\mathrm{x}_{\mathrm{12}} ,\:\mathrm{ar}^{\mathrm{3}} ,...\:. \\ $$$$\mathrm{where}\::\:\mathrm{a},\mathrm{x}_{\mathrm{1}} ,\mathrm{x}_{\mathrm{2}} ,\mathrm{ar}\:\Rightarrow\mathrm{AP} \\ $$$$\mathrm{ar},\mathrm{x}_{\mathrm{3}} ,\mathrm{x}_{\mathrm{4}} ,\mathrm{x}_{\mathrm{5}} ,\mathrm{x}_{\mathrm{6}} ,\mathrm{ar}^{\mathrm{2}} \Rightarrow\mathrm{AP} \\ $$$$\mathrm{ar}^{\mathrm{2}} ,\mathrm{x}_{\mathrm{7}} ,\mathrm{x}_{\mathrm{8}} ,\mathrm{x}_{\mathrm{9}} ,\mathrm{x}_{\mathrm{10}} ,\mathrm{x}_{\mathrm{11}} ,\mathrm{x}_{\mathrm{12}} ,\mathrm{ar}^{\mathrm{3}} \Rightarrow\mathrm{AP} \\ $$$$...\mathrm{etc} \\ $$$$\mathrm{if}\:\underset{\mathrm{n}\rightarrow\infty} {\mathrm{lim}}\:\left(\mathrm{x}_{\mathrm{1}} +\mathrm{x}_{\mathrm{2}} +\mathrm{x}_{\mathrm{3}} +...\right)=\:\frac{\mathrm{21}}{\mathrm{16}}×\frac{\mathrm{a}}{\mathrm{1}−\mathrm{r}} \\ $$$$\mathrm{what}\:\mathrm{is}\:\mathrm{r}\:? \\ $$

Question Number 79647    Answers: 1   Comments: 2

discussion back with mr W. consider this equation (x^2 −2x−3)(3^x −27)=0 does the equation have two roots or three roots?

$$\mathrm{discussion}\:\mathrm{back}\:\mathrm{with}\:\mathrm{mr}\:\mathrm{W}. \\ $$$$\mathrm{consider}\:\mathrm{this}\:\mathrm{equation}\: \\ $$$$\left(\mathrm{x}^{\mathrm{2}} −\mathrm{2x}−\mathrm{3}\right)\left(\mathrm{3}^{\mathrm{x}} −\mathrm{27}\right)=\mathrm{0} \\ $$$$\mathrm{does}\:\mathrm{the}\:\mathrm{equation}\:\mathrm{have}\:\mathrm{two}\:\mathrm{roots} \\ $$$$\mathrm{or}\:\mathrm{three}\:\mathrm{roots}? \\ $$

Question Number 79646    Answers: 0   Comments: 1

calculate A_n =∫_0 ^1 cos(narcosx)dx with n integr natural

$${calculate}\:\:{A}_{{n}} =\int_{\mathrm{0}} ^{\mathrm{1}} \:{cos}\left({narcosx}\right){dx} \\ $$$${with}\:{n}\:{integr}\:{natural} \\ $$

Question Number 79645    Answers: 0   Comments: 0

find ∫_0 ^1 ln(1+x^4 )dx

$${find}\:\int_{\mathrm{0}} ^{\mathrm{1}} \:{ln}\left(\mathrm{1}+{x}^{\mathrm{4}} \right){dx} \\ $$

Question Number 79644    Answers: 1   Comments: 1

f(x) = ((4(4x^2 +3))/(4x^2 +4x+5)) prove that f(x) ≥ 2 .

$${f}\left({x}\right)\:=\:\frac{\mathrm{4}\left(\mathrm{4}{x}^{\mathrm{2}} +\mathrm{3}\right)}{\mathrm{4}{x}^{\mathrm{2}} +\mathrm{4}{x}+\mathrm{5}} \\ $$$$\mathrm{prove}\:\mathrm{that}\:{f}\left({x}\right)\:\geqslant\:\mathrm{2}\:. \\ $$

Question Number 79635    Answers: 0   Comments: 9

Sum: (1/1) + (1/(1 + 2)) + (1/(1 + 2 + 3)) + ... + (1/(1 + 2 + 3 + ... + 8016))

$$\mathrm{Sum}:\:\:\frac{\mathrm{1}}{\mathrm{1}}\:+\:\frac{\mathrm{1}}{\mathrm{1}\:+\:\mathrm{2}}\:+\:\frac{\mathrm{1}}{\mathrm{1}\:+\:\mathrm{2}\:+\:\mathrm{3}}\:+\:...\:\:+\:\:\frac{\mathrm{1}}{\mathrm{1}\:+\:\mathrm{2}\:+\:\mathrm{3}\:+\:...\:+\:\mathrm{8016}} \\ $$

Question Number 79634    Answers: 1   Comments: 3

Question Number 79627    Answers: 0   Comments: 2

1) expicite f(x)=∫_0 ^1 ((ln(1+xt^2 ))/(1+t^2 ))dt with x≥0 2)calculate ∫_0 ^1 ((ln(1+t^2 ))/(1+t^2 ))dt and ∫_0 ^1 ((ln(1+2t^2 ))/(1+t^2 ))dt

$$\left.\mathrm{1}\right)\:{expicite}\:{f}\left({x}\right)=\int_{\mathrm{0}} ^{\mathrm{1}} \:\frac{{ln}\left(\mathrm{1}+{xt}^{\mathrm{2}} \right)}{\mathrm{1}+{t}^{\mathrm{2}} }{dt}\:{with}\:{x}\geqslant\mathrm{0} \\ $$$$\left.\mathrm{2}\right){calculate}\:\int_{\mathrm{0}} ^{\mathrm{1}} \:\frac{{ln}\left(\mathrm{1}+{t}^{\mathrm{2}} \right)}{\mathrm{1}+{t}^{\mathrm{2}} }{dt}\:{and}\:\int_{\mathrm{0}} ^{\mathrm{1}} \:\frac{{ln}\left(\mathrm{1}+\mathrm{2}{t}^{\mathrm{2}} \right)}{\mathrm{1}+{t}^{\mathrm{2}} }{dt} \\ $$

Question Number 79626    Answers: 1   Comments: 0

find tbe general solution of the D.E y^(′′) −5y^′ +4y=e^t cos(t)

$${find}\:{tbe}\:{general}\:{solution}\:{of}\:{the}\:{D}.{E} \\ $$$${y}^{''} −\mathrm{5}{y}^{'} +\mathrm{4}{y}={e}^{{t}} {cos}\left({t}\right) \\ $$

Question Number 79625    Answers: 1   Comments: 0

if n>1 prove that 2ln(n)−ln(n+1)−ln(n−1)=(1/n^2 )+(1/(2n^4 ))+(1/(3n^6 ))+...=

$${if}\:{n}>\mathrm{1}\:{prove}\:{that} \\ $$$$\mathrm{2}{ln}\left({n}\right)−{ln}\left({n}+\mathrm{1}\right)−{ln}\left({n}−\mathrm{1}\right)=\frac{\mathrm{1}}{{n}^{\mathrm{2}} }+\frac{\mathrm{1}}{\mathrm{2}{n}^{\mathrm{4}} }+\frac{\mathrm{1}}{\mathrm{3}{n}^{\mathrm{6}} }+...= \\ $$

Question Number 79615    Answers: 0   Comments: 3

prove that with using hypergeometric function ∫_0 ^π sin(x^2 )=(π^3 /3) 1F_2 [(3/4);(3/2);(7/4);((−π^4 )/4)]

$${prove}\:{that}\:{with}\:{using}\:{hypergeometric}\:{function} \\ $$$$\int_{\mathrm{0}} ^{\pi} {sin}\left({x}^{\mathrm{2}} \right)=\frac{\pi^{\mathrm{3}} }{\mathrm{3}}\:\mathrm{1}{F}_{\mathrm{2}} \left[\frac{\mathrm{3}}{\mathrm{4}};\frac{\mathrm{3}}{\mathrm{2}};\frac{\mathrm{7}}{\mathrm{4}};\frac{−\pi^{\mathrm{4}} }{\mathrm{4}}\right]\: \\ $$

Question Number 79609    Answers: 1   Comments: 1

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