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Question Number 67018 Answers: 0 Comments: 1

find ∫_0 ^∞ e^(−x) ln(1+x)dx

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

Question Number 67017 Answers: 0 Comments: 1

find ∫ arctan(1+(√(x+1)))dx

$${find}\:\int\:\:{arctan}\left(\mathrm{1}+\sqrt{{x}+\mathrm{1}}\right){dx} \\ $$

Question Number 67016 Answers: 0 Comments: 0

find ∫ arctan(1+(√x)+(√(x+1)))dx

$${find}\:\int\:{arctan}\left(\mathrm{1}+\sqrt{{x}}+\sqrt{{x}+\mathrm{1}}\right){dx} \\ $$

Question Number 67015 Answers: 0 Comments: 0

let f(x) =arctan(1+e^(−(√(1+x^2 ))) ) calculate f^′ (x) and f^(′′) (x). 1)find lim_(x→+∞) f(x) and lim_(x→−∞) f(x) 3)study the variation of f(x) 4)give the equation of tangent to C_f at A(1,f(1))

$${let}\:{f}\left({x}\right)\:={arctan}\left(\mathrm{1}+{e}^{−\sqrt{\mathrm{1}+{x}^{\mathrm{2}} }} \right) \\ $$$${calculate}\:{f}^{'} \left({x}\right)\:\:{and}\:{f}^{''} \left({x}\right). \\ $$$$\left.\mathrm{1}\right){find}\:{lim}_{{x}\rightarrow+\infty} {f}\left({x}\right)\:{and}\:{lim}_{{x}\rightarrow−\infty} \:\:\:{f}\left({x}\right) \\ $$$$\left.\mathrm{3}\right){study}\:{the}\:{variation}\:{of}\:{f}\left({x}\right) \\ $$$$\left.\mathrm{4}\right){give}\:{the}\:{equation}\:{of}\:{tangent}\:{to}\:{C}_{{f}} \:{at}\:\:{A}\left(\mathrm{1},{f}\left(\mathrm{1}\right)\right) \\ $$

Question Number 67014 Answers: 0 Comments: 1

solve y^(′′) +x^2 y^′ =e^(−x) sin(3x)

$${solve}\:{y}^{''} +{x}^{\mathrm{2}} {y}^{'} ={e}^{−{x}} {sin}\left(\mathrm{3}{x}\right) \\ $$

Question Number 67013 Answers: 0 Comments: 1

find the value of Σ_(n=1) ^∞ (((−1)^n )/((n+1)n^3 ))

$${find}\:{the}\:{value}\:{of}\:\sum_{{n}=\mathrm{1}} ^{\infty} \:\:\:\frac{\left(−\mathrm{1}\right)^{{n}} }{\left({n}+\mathrm{1}\right){n}^{\mathrm{3}} } \\ $$

Question Number 67012 Answers: 0 Comments: 1

find ∫_1 ^(+∞) ((arctan([x]))/x^3 )dx

$${find}\:\int_{\mathrm{1}} ^{+\infty} \:\frac{{arctan}\left(\left[{x}\right]\right)}{{x}^{\mathrm{3}} }{dx} \\ $$

Question Number 67011 Answers: 0 Comments: 1

calculate U_n =∫_1 ^(+∞) ((arctan(n[x]))/x^2 )dx

$${calculate}\:{U}_{{n}} =\int_{\mathrm{1}} ^{+\infty} \:\:\frac{{arctan}\left({n}\left[{x}\right]\right)}{{x}^{\mathrm{2}} }{dx} \\ $$

Question Number 67010 Answers: 0 Comments: 1

calculate Σ_(n=4) ^(+∞) (n/((n^2 −9)^2 ))

$${calculate}\:\:\sum_{{n}=\mathrm{4}} ^{+\infty} \:\:\:\:\frac{{n}}{\left({n}^{\mathrm{2}} −\mathrm{9}\right)^{\mathrm{2}} } \\ $$

Question Number 67008 Answers: 0 Comments: 2

let f(x) =∫_0 ^∞ (dt/((x^2 +t^2 )^2 )) with x>0 1) find a explicit form of (x) 2)find also g(x) =∫_0 ^∞ (dt/((x^2 +t^2 )^3 )) 3)find the values of integrals ∫_0 ^∞ (dt/((t^2 +3)^2 )) and ∫_0 ^∞ (dt/((t^2 +3)^3 )) 4) calculate U_θ =∫_0 ^∞ (dt/((t^2 +cos^2 θ)^2 )) with 0<θ<(π/2) 5) find f^((n)) (x) and f^((n)) (0) 6) developp f at integr serie

$${let}\:{f}\left({x}\right)\:=\int_{\mathrm{0}} ^{\infty} \:\:\:\:\frac{{dt}}{\left({x}^{\mathrm{2}} \:+{t}^{\mathrm{2}} \right)^{\mathrm{2}} }\:\:{with}\:{x}>\mathrm{0} \\ $$$$\left.\mathrm{1}\right)\:{find}\:{a}\:{explicit}\:{form}\:{of}\:\left({x}\right) \\ $$$$\left.\mathrm{2}\right){find}\:{also}\:{g}\left({x}\right)\:=\int_{\mathrm{0}} ^{\infty} \:\:\:\frac{{dt}}{\left({x}^{\mathrm{2}} \:+{t}^{\mathrm{2}} \right)^{\mathrm{3}} } \\ $$$$\left.\mathrm{3}\right){find}\:{the}\:{values}\:{of}\:{integrals}\:\int_{\mathrm{0}} ^{\infty} \:\:\:\frac{{dt}}{\left({t}^{\mathrm{2}} \:+\mathrm{3}\right)^{\mathrm{2}} }\:{and}\:\int_{\mathrm{0}} ^{\infty} \:\:\frac{{dt}}{\left({t}^{\mathrm{2}} \:+\mathrm{3}\right)^{\mathrm{3}} } \\ $$$$\left.\mathrm{4}\right)\:{calculate}\:{U}_{\theta} =\int_{\mathrm{0}} ^{\infty} \:\:\:\frac{{dt}}{\left({t}^{\mathrm{2}} \:+{cos}^{\mathrm{2}} \theta\right)^{\mathrm{2}} }\:\:\:{with}\:\mathrm{0}<\theta<\frac{\pi}{\mathrm{2}} \\ $$$$\left.\mathrm{5}\right)\:{find}\:{f}^{\left({n}\right)} \left({x}\right)\:{and}\:{f}^{\left({n}\right)} \left(\mathrm{0}\right) \\ $$$$\left.\mathrm{6}\right)\:{developp}\:{f}\:{at}\:{integr}\:{serie} \\ $$

Question Number 67006 Answers: 0 Comments: 1

calculae A_n =∫_0 ^∞ (dx/((x^2 +1)^n )) with n integr natural and n>0

$${calculae}\:{A}_{{n}} =\int_{\mathrm{0}} ^{\infty} \:\:\:\:\frac{{dx}}{\left({x}^{\mathrm{2}} +\mathrm{1}\right)^{{n}} }\:\:\:{with}\:{n}\:{integr}\:{natural}\:{and}\:{n}>\mathrm{0} \\ $$

Question Number 67005 Answers: 0 Comments: 1

find ∫ ((x−2(√(x^2 −1)))/(x+2(√(x^2 −1))))dx

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

Question Number 67004 Answers: 1 Comments: 0

Question Number 66995 Answers: 0 Comments: 4

Question Number 66988 Answers: 0 Comments: 0

Question Number 66986 Answers: 0 Comments: 1

Question Number 66985 Answers: 1 Comments: 0

The external length,width and height of an open rectangular container are 41cm,21cm and 15.5cm respectively. The thickness of the material making the container is 5mm.If the container has 8litres of water,calculate the internal height above the water level. Ans:5cm

$$\mathrm{The}\:\mathrm{external}\:\mathrm{length},\mathrm{width}\:\mathrm{and}\:\mathrm{height} \\ $$$$\mathrm{of}\:\mathrm{an}\:\mathrm{open}\:\mathrm{rectangular}\:\mathrm{container}\:\mathrm{are} \\ $$$$\mathrm{41cm},\mathrm{21cm}\:\mathrm{and}\:\mathrm{15}.\mathrm{5cm}\:\mathrm{respectively}. \\ $$$$\mathrm{The}\:\mathrm{thickness}\:\mathrm{of}\:\mathrm{the}\:\mathrm{material}\:\mathrm{making} \\ $$$$\mathrm{the}\:\mathrm{container}\:\mathrm{is}\:\mathrm{5mm}.\mathrm{If}\:\mathrm{the}\:\mathrm{container} \\ $$$$\mathrm{has}\:\mathrm{8litres}\:\mathrm{of}\:\mathrm{water},\mathrm{calculate}\:\mathrm{the} \\ $$$$\mathrm{internal}\:\mathrm{height}\:\mathrm{above}\:\mathrm{the}\:\mathrm{water}\:\mathrm{level}. \\ $$$$\:\:\:\:\:\:\:\: \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:{Ans}:\mathrm{5}{cm} \\ $$

Question Number 66981 Answers: 1 Comments: 1

Question Number 66980 Answers: 0 Comments: 1

Question Number 66979 Answers: 0 Comments: 1

Question Number 66959 Answers: 0 Comments: 0

Question Number 66938 Answers: 2 Comments: 7

Question Number 66937 Answers: 1 Comments: 0

Question Number 66932 Answers: 1 Comments: 0

which of the sequences converges ? 1) a_n = n−(1/n) 2) a_n = ((−1)^n +1)(((n+1)/n))

$${which}\:{of}\:{the}\:{sequences}\:{converges}\:? \\ $$$$\left.\mathrm{1}\right)\:{a}_{{n}} =\:{n}−\frac{\mathrm{1}}{{n}} \\ $$$$\left.\mathrm{2}\right)\:{a}_{{n}} =\:\left(\left(−\mathrm{1}\right)^{{n}} +\mathrm{1}\right)\left(\frac{{n}+\mathrm{1}}{{n}}\right) \\ $$

Question Number 66928 Answers: 2 Comments: 0

(3^(120) /6^(60) ) express in your answer in 3 significant figure

$$\frac{\mathrm{3}^{\mathrm{120}} }{\mathrm{6}^{\mathrm{60}} } \\ $$$$\mathrm{express}\:\mathrm{in}\:\mathrm{your}\:\mathrm{answer}\:\mathrm{in} \\ $$$$\mathrm{3}\:\mathrm{significant}\:\mathrm{figure} \\ $$

Question Number 66927 Answers: 0 Comments: 0

calcul la limite suivante: lim (((1^x +2^x +3^x +......+n^x )/n))^(1/x) =A x→0 trouve la valeur de A trouve la valeur de P definir par: P=(A/((((n! ))^(1/n) )^((n−1)) ))=???

$${calcul}\:{la}\:{limite}\:{suivante}: \\ $$$${lim}\:\:\:\:\:\:\:\left(\frac{\mathrm{1}^{{x}} +\mathrm{2}^{{x}} +\mathrm{3}^{{x}} +......+{n}^{{x}} }{{n}}\right)^{\frac{\mathrm{1}}{{x}}} =\mathrm{A} \\ $$$${x}\rightarrow\mathrm{0} \\ $$$$\mathrm{trouve}\:\mathrm{la}\:\mathrm{valeur}\:\mathrm{de}\:\boldsymbol{\mathrm{A}} \\ $$$$\mathrm{trouve}\:\mathrm{la}\:\mathrm{valeur}\:\mathrm{de}\:\boldsymbol{\mathrm{P}}\:\mathrm{definir}\:\mathrm{par}: \\ $$$$\boldsymbol{\mathrm{P}}=\frac{\boldsymbol{\mathrm{A}}}{\left(\sqrt[{\mathrm{n}}]{\mathrm{n}!\:}\right)^{\left(\mathrm{n}−\mathrm{1}\right)} }=??? \\ $$

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