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Question Number 56386    Answers: 0   Comments: 0

Question Number 56384    Answers: 1   Comments: 2

Question Number 56383    Answers: 2   Comments: 4

Question Number 56378    Answers: 0   Comments: 2

Question Number 56368    Answers: 1   Comments: 2

What is the common formula to obtain the 3 solutions of a polynomial equation in the following form ? ax^3 + b^2 x + cx + d = 0

$${What}\:{is}\:{the}\:{common}\:{formula}\:{to}\:{obtain} \\ $$$${the}\:\mathrm{3}\:{solutions}\:{of}\:{a}\:{polynomial}\:{equation} \\ $$$${in}\:{the}\:{following}\:{form}\:? \\ $$$$ \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:{ax}^{\mathrm{3}} \:+\:{b}^{\mathrm{2}} {x}\:+\:{cx}\:+\:{d}\:=\:\mathrm{0} \\ $$

Question Number 56358    Answers: 0   Comments: 3

∫_0 ^∞ (cot^(−1) x)^2 dx

$$\int_{\mathrm{0}} ^{\infty} \left(\mathrm{cot}^{−\mathrm{1}} {x}\right)^{\mathrm{2}} {dx} \\ $$$$ \\ $$

Question Number 56356    Answers: 3   Comments: 1

Find the minimum value of the function F(x) = log_e x − x for x > 0. Hence show that: log_e x ≤ x − 1. For all x > 0

$$\mathrm{Find}\:\mathrm{the}\:\mathrm{minimum}\:\mathrm{value}\:\mathrm{of}\:\mathrm{the}\:\mathrm{function}\:\mathrm{F}\left(\mathrm{x}\right)\:=\:\:\mathrm{log}_{\mathrm{e}} \mathrm{x}\:\:−\:\:\mathrm{x}\:\:\:\mathrm{for}\:\:\:\mathrm{x}\:>\:\mathrm{0}. \\ $$$$\:\:\mathrm{Hence}\:\mathrm{show}\:\mathrm{that}:\:\:\:\mathrm{log}_{\mathrm{e}} \mathrm{x}\:\:\leqslant\:\:\mathrm{x}\:−\:\mathrm{1}.\:\:\mathrm{For}\:\mathrm{all}\:\:\:\mathrm{x}\:>\:\mathrm{0} \\ $$

Question Number 56351    Answers: 1   Comments: 0

log_3 (2y+1)+2=log_3 6y^2 −log_3 2y

$$\mathrm{log}_{\mathrm{3}} \left(\mathrm{2y}+\mathrm{1}\right)+\mathrm{2}=\mathrm{log}_{\mathrm{3}} \mathrm{6y}^{\mathrm{2}} −\mathrm{log}_{\mathrm{3}} \mathrm{2y} \\ $$

Question Number 56339    Answers: 1   Comments: 0

Question Number 56336    Answers: 1   Comments: 0

make r the subject of P=(1+(r/(100)))^t

$$\mathrm{make}\:\:\mathrm{r}\:\:\mathrm{the}\:\mathrm{subject}\:\mathrm{of}\:\:\: \\ $$$$\mathrm{P}=\left(\mathrm{1}+\frac{\mathrm{r}}{\mathrm{100}}\right)^{\mathrm{t}} \\ $$

Question Number 56335    Answers: 2   Comments: 0

If b^x = ((b/k))^y = k^m and b ≠ 1 show that: (1/x) = (1/y) = (1/m)

$$\mathrm{If}\:\:\:\:\:\:\mathrm{b}^{\mathrm{x}} \:\:=\:\:\left(\frac{\mathrm{b}}{\mathrm{k}}\right)^{\mathrm{y}} \:\:=\:\:\mathrm{k}^{\mathrm{m}} \:\:\:\:\:\:\:\:\:\mathrm{and}\:\:\:\:\:\:\mathrm{b}\:\neq\:\mathrm{1} \\ $$$$\mathrm{show}\:\mathrm{that}:\:\:\:\:\:\:\frac{\mathrm{1}}{\mathrm{x}}\:=\:\frac{\mathrm{1}}{\mathrm{y}}\:=\:\frac{\mathrm{1}}{\mathrm{m}} \\ $$

Question Number 56332    Answers: 1   Comments: 0

calculate lim_(x→0) ((arctan(1+cosx)−arctan(2+x))/x^3 )

$${calculate}\:{lim}_{{x}\rightarrow\mathrm{0}} \:\:\:\frac{{arctan}\left(\mathrm{1}+{cosx}\right)−{arctan}\left(\mathrm{2}+{x}\right)}{{x}^{\mathrm{3}} } \\ $$

Question Number 56331    Answers: 0   Comments: 0

calculate lim_(x→0) ((ch(cosx)−cos(chx))/x^2 )

$${calculate}\:{lim}_{{x}\rightarrow\mathrm{0}} \:\:\:\:\:\frac{{ch}\left({cosx}\right)−{cos}\left({chx}\right)}{{x}^{\mathrm{2}} } \\ $$

Question Number 56330    Answers: 0   Comments: 1

calculate Σ_(n=0) ^∞ (((−1)^n )/(n^2 (3n+1)))

$${calculate}\:\sum_{{n}=\mathrm{0}} ^{\infty} \:\:\:\frac{\left(−\mathrm{1}\right)^{{n}} }{{n}^{\mathrm{2}} \left(\mathrm{3}{n}+\mathrm{1}\right)} \\ $$

Question Number 56329    Answers: 0   Comments: 1

1)calculate A_n =∫_(1/n) ^1 ((ln(1+x^2 ))/(1+x^2 ))dx with n integr and n≥1 2) find lim_(n→+∞) A_n 3) study the convergence of Σ A_n

$$\left.\mathrm{1}\right){calculate}\:{A}_{{n}} =\int_{\frac{\mathrm{1}}{{n}}} ^{\mathrm{1}} \:\:\frac{{ln}\left(\mathrm{1}+{x}^{\mathrm{2}} \right)}{\mathrm{1}+{x}^{\mathrm{2}} }{dx}\:\:\:\:{with}\:{n}\:{integr}\:{and}\:{n}\geqslant\mathrm{1} \\ $$$$\left.\mathrm{2}\right)\:{find}\:{lim}_{{n}\rightarrow+\infty} \:\:\:\:{A}_{{n}} \\ $$$$\left.\mathrm{3}\right)\:\:{study}\:{the}\:{convergence}\:{of}\:\Sigma\:{A}_{{n}} \\ $$

Question Number 56345    Answers: 0   Comments: 1

let f(a) =∫_0 ^∞ (dx/(x^n +a^n )) with n integr ≥2 and a>0 1) calculate f(a) intems of a 2) let g(a) =∫_0 ^∞ (dx/((x^n +a^n )^2 )) calculate g(a) interms of a 3) find the values of integrals ∫_0 ^∞ (dx/(x^8 +16)) and ∫_0 ^∞ (dx/((x^8 +16)^2 ))

$${let}\:\:{f}\left({a}\right)\:=\int_{\mathrm{0}} ^{\infty} \:\:\:\:\:\frac{{dx}}{{x}^{{n}} \:+{a}^{{n}} }\:\:\:{with}\:{n}\:{integr}\:\geqslant\mathrm{2}\:\:{and}\:{a}>\mathrm{0} \\ $$$$\left.\mathrm{1}\right)\:{calculate}\:{f}\left({a}\right)\:{intems}\:{of}\:{a} \\ $$$$\left.\mathrm{2}\right)\:{let}\:{g}\left({a}\right)\:=\int_{\mathrm{0}} ^{\infty} \:\:\:\frac{{dx}}{\left({x}^{{n}} \:+{a}^{{n}} \right)^{\mathrm{2}} }\:\:{calculate}\:{g}\left({a}\right)\:{interms}\:{of}\:{a} \\ $$$$\left.\mathrm{3}\right)\:{find}\:{the}\:{values}\:{of}\:{integrals}\:\int_{\mathrm{0}} ^{\infty} \:\:\frac{{dx}}{{x}^{\mathrm{8}} +\mathrm{16}}\:\:\:{and}\:\int_{\mathrm{0}} ^{\infty} \:\:\:\frac{{dx}}{\left({x}^{\mathrm{8}} \:+\mathrm{16}\right)^{\mathrm{2}} } \\ $$

Question Number 56500    Answers: 1   Comments: 1

find the X_0 value of the x variable with which the function assumes the lower value. f(x)=x^2 −3x+7+sin(πx)

$$\boldsymbol{{find}}\:\boldsymbol{{the}}\:\boldsymbol{{X}}_{\mathrm{0}} \:\boldsymbol{{value}}\:\boldsymbol{{of}}\:\boldsymbol{{the}}\:\boldsymbol{{x}}\:\boldsymbol{{variable}}\:\boldsymbol{{with}}\:\boldsymbol{{which}} \\ $$$$\boldsymbol{{the}}\:\boldsymbol{{function}}\:\boldsymbol{{assumes}}\:\boldsymbol{{the}}\:\boldsymbol{{lower}}\:\boldsymbol{{value}}. \\ $$$$ \\ $$$${f}\left({x}\right)={x}^{\mathrm{2}} −\mathrm{3}{x}+\mathrm{7}+{sin}\left(\pi{x}\right) \\ $$

Question Number 56503    Answers: 1   Comments: 0

A boat travels 30km upstream and 44km downstream in 10 hours. in 13 hours it can travel 40km upstream and 55km downstream. Determine the speed of the stream and that of the boat in still water. (in km/hr)

$$\mathrm{A}\:\mathrm{boat}\:\mathrm{travels}\:\mathrm{30km}\:\mathrm{upstream}\:\mathrm{and}\: \\ $$$$\mathrm{44km}\:\mathrm{downstream}\:\mathrm{in}\:\mathrm{10}\:\mathrm{hours}.\: \\ $$$$\mathrm{in}\:\mathrm{13}\:\mathrm{hours}\:\mathrm{it}\:\mathrm{can}\:\mathrm{travel}\:\mathrm{40km}\:\mathrm{upstream} \\ $$$$\mathrm{and}\:\mathrm{55km}\:\mathrm{downstream}.\:\mathrm{Determine}\:\mathrm{the} \\ $$$$\mathrm{speed}\:\mathrm{of}\:\mathrm{the}\:\mathrm{stream}\:\:\mathrm{and}\:\mathrm{that}\:\mathrm{of}\:\mathrm{the}\: \\ $$$$\mathrm{boat}\:\mathrm{in}\:\mathrm{still}\:\mathrm{water}.\:\left(\mathrm{in}\:\mathrm{km}/\mathrm{hr}\right) \\ $$

Question Number 56325    Answers: 0   Comments: 3

Happy π−day

$${Happy}\:\pi−{day} \\ $$

Question Number 56311    Answers: 0   Comments: 1

let f(x) =∫_0 ^∞ ((cos(xt))/(x^2 +t^2 )) dt with x>0 1) find f(x) 2) find the values of ∫_0 ^∞ ((cos(t))/(1+t^2 ))dt and ∫_0 ^∞ ((cos(2t))/(4+t^2 ))dt 3) let U_n =∫_0 ^∞ ((cos(nt))/(n^2 +t^2 ))dt find lim_(n→+∞) U_n and study the convergenge of Σ U_n and Σ U_n ^2

$${let}\:{f}\left({x}\right)\:=\int_{\mathrm{0}} ^{\infty} \:\:\frac{{cos}\left({xt}\right)}{{x}^{\mathrm{2}} \:+{t}^{\mathrm{2}} }\:{dt}\:\:{with}\:{x}>\mathrm{0} \\ $$$$\left.\mathrm{1}\right)\:{find}\:{f}\left({x}\right) \\ $$$$\left.\mathrm{2}\right)\:{find}\:{the}\:{values}\:{of}\:\int_{\mathrm{0}} ^{\infty} \:\:\frac{{cos}\left({t}\right)}{\mathrm{1}+{t}^{\mathrm{2}} }{dt}\:{and}\:\int_{\mathrm{0}} ^{\infty} \:\frac{{cos}\left(\mathrm{2}{t}\right)}{\mathrm{4}+{t}^{\mathrm{2}} }{dt} \\ $$$$\left.\mathrm{3}\right)\:{let}\:{U}_{{n}} =\int_{\mathrm{0}} ^{\infty} \:\:\frac{{cos}\left({nt}\right)}{{n}^{\mathrm{2}} +{t}^{\mathrm{2}} }{dt}\:\:\:{find}\:{lim}_{{n}\rightarrow+\infty} {U}_{{n}} \:\:\:\:{and}\:{study}\:{the}\:{convergenge}\:{of} \\ $$$$\Sigma\:{U}_{{n}} \:\:\:{and}\:\Sigma\:{U}_{{n}} ^{\mathrm{2}} \\ $$$$ \\ $$

Question Number 56310    Answers: 0   Comments: 2

let f(x)=∫_(−∞) ^(+∞) cos(t^2 +xt +3)dt with x>0 1) find f(x) 2) calculate ∫_1 ^4 f(x)dx and ∫_1 ^(+∞) f(x)dx

$${let}\:{f}\left({x}\right)=\int_{−\infty} ^{+\infty} \:\:{cos}\left({t}^{\mathrm{2}} \:+{xt}\:+\mathrm{3}\right){dt}\:\:{with}\:{x}>\mathrm{0} \\ $$$$\left.\mathrm{1}\right)\:{find}\:{f}\left({x}\right) \\ $$$$\left.\mathrm{2}\right)\:{calculate}\:\int_{\mathrm{1}} ^{\mathrm{4}} {f}\left({x}\right){dx}\:{and}\:\int_{\mathrm{1}} ^{+\infty} {f}\left({x}\right){dx} \\ $$

Question Number 56321    Answers: 2   Comments: 0

Solve for x and y x (√x) + y(√y) = 182 ..... (i) x (√y) + y(√x) = 183 ..... (ii)

$$\mathrm{Solve}\:\mathrm{for}\:\:\mathrm{x}\:\mathrm{and}\:\mathrm{y} \\ $$$$\:\:\:\:\:\:\mathrm{x}\:\sqrt{\mathrm{x}}\:\:+\:\mathrm{y}\sqrt{\mathrm{y}}\:\:=\:\mathrm{182}\:\:\:\:\:\:.....\:\left(\mathrm{i}\right) \\ $$$$\:\:\:\:\:\:\mathrm{x}\:\sqrt{\mathrm{y}}\:\:+\:\mathrm{y}\sqrt{\mathrm{x}}\:\:=\:\mathrm{183}\:\:\:\:\:\:.....\:\left(\mathrm{ii}\right) \\ $$

Question Number 56316    Answers: 1   Comments: 1

Question Number 56301    Answers: 2   Comments: 1

Minimum value of b that satisfy the following inequality ((53)/(201)) < (a/b) < (4/(15)) is ...

$${Minimum}\:\:{value}\:\:{of}\:\:{b}\:\:{that}\:\:{satisfy}\:\:{the} \\ $$$${following}\:\:{inequality}\:\: \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\frac{\mathrm{53}}{\mathrm{201}}\:\:<\:\:\frac{{a}}{{b}}\:\:<\:\:\frac{\mathrm{4}}{\mathrm{15}}\:\:\:\:\:{is}\:\:\:... \\ $$

Question Number 56289    Answers: 1   Comments: 7

Question Number 56282    Answers: 0   Comments: 5

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