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Question Number 34226 Answers: 0 Comments: 1

let u_n = (n+1)^((n+1)/n) −n^(n/(n−1)) find lim_(n→+∞) u_n

$${let}\:{u}_{{n}} =\:\left({n}+\mathrm{1}\right)^{\frac{{n}+\mathrm{1}}{{n}}} \:\:−{n}^{\frac{{n}}{{n}−\mathrm{1}}} \\ $$$${find}\:{lim}_{{n}\rightarrow+\infty} {u}_{{n}} \\ $$

Question Number 34225 Answers: 1 Comments: 0

find ∫ (dx/(1+x^2 +x^4 ))

$${find}\:\int\:\:\:\frac{{dx}}{\mathrm{1}+{x}^{\mathrm{2}} +{x}^{\mathrm{4}} } \\ $$

Question Number 34224 Answers: 0 Comments: 0

study and give the graph for f(x) =(((x−1)^2 )/x) e^(−(1/(x^2 ))) .

$${study}\:{and}\:{give}\:{the}\:{graph}\:\:{for} \\ $$$${f}\left({x}\right)\:=\frac{\left({x}−\mathrm{1}\right)^{\mathrm{2}} }{{x}}\:{e}^{−\frac{\mathrm{1}}{{x}^{\mathrm{2}} \:\:}} . \\ $$

Question Number 34223 Answers: 0 Comments: 0

find ∫ (dx/(x^(2n) −1)) with n integr natural and n≥1 .

$${find}\:\int\:\:\:\frac{{dx}}{{x}^{\mathrm{2}{n}} −\mathrm{1}}\:\:{with}\:{n}\:{integr}\:{natural}\:{and}\:{n}\geqslant\mathrm{1}\:. \\ $$

Question Number 34222 Answers: 0 Comments: 4

let give the sequence of integrals J_n =∫_0 ^∞ x^n e^(−(x^2 /2)) dx 1) prove that J_n =(n−1)J_(n−2) ∀n≥2 2) calculate J_(2p) and J_(2p+1) by using factoriels. 3) prove that ∀n≥1 J_n ^2 ≤J_(n−1) . J_(n+1) . 4)prove that ((2^(2p) (p!)^2 )/((2p)!)) (1/(√(2p+1))) ≤J_0 ≤ ((2^(2p) (p!)^2 )/((2p)!)) (1/(√(2p))) 5) find a equivalent of ((2^(2p) (p!)^2 )/((2p)!)) (p→+∞)

$${let}\:{give}\:{the}\:{sequence}\:{of}\:{integrals} \\ $$$${J}_{{n}} =\int_{\mathrm{0}} ^{\infty} \:{x}^{{n}} \:\:{e}^{−\frac{{x}^{\mathrm{2}} }{\mathrm{2}}} {dx} \\ $$$$\left.\mathrm{1}\right)\:{prove}\:{that}\:{J}_{{n}} =\left({n}−\mathrm{1}\right){J}_{{n}−\mathrm{2}} \:\:\:\forall{n}\geqslant\mathrm{2} \\ $$$$\left.\mathrm{2}\right)\:{calculate}\:{J}_{\mathrm{2}{p}} \:{and}\:{J}_{\mathrm{2}{p}+\mathrm{1}} \:{by}\:{using}\:{factoriels}. \\ $$$$\left.\mathrm{3}\right)\:{prove}\:{that}\:\:\forall{n}\geqslant\mathrm{1}\:\:\:{J}_{{n}} ^{\mathrm{2}} \:\:\leqslant{J}_{{n}−\mathrm{1}} \:.\:{J}_{{n}+\mathrm{1}} . \\ $$$$\left.\mathrm{4}\right){prove}\:{that}\:\:\frac{\mathrm{2}^{\mathrm{2}{p}} \left({p}!\right)^{\mathrm{2}} }{\left(\mathrm{2}{p}\right)!}\:\frac{\mathrm{1}}{\sqrt{\mathrm{2}{p}+\mathrm{1}}}\:\leqslant{J}_{\mathrm{0}} \:\leqslant\:\frac{\mathrm{2}^{\mathrm{2}{p}} \:\left({p}!\right)^{\mathrm{2}} }{\left(\mathrm{2}{p}\right)!}\:\frac{\mathrm{1}}{\sqrt{\mathrm{2}{p}}} \\ $$$$\left.\mathrm{5}\right)\:{find}\:{a}\:{equivalent}\:{of}\:\:\frac{\mathrm{2}^{\mathrm{2}{p}} \left({p}!\right)^{\mathrm{2}} }{\left(\mathrm{2}{p}\right)!}\:\:\left({p}\rightarrow+\infty\right) \\ $$

Question Number 34221 Answers: 1 Comments: 1

study the convergence of ∫_0 ^1 ((√(1−x))/x) dx .

$${study}\:{the}\:{convergence}\:{of}\:\int_{\mathrm{0}} ^{\mathrm{1}} \:\:\frac{\sqrt{\mathrm{1}−{x}}}{{x}}\:{dx}\:. \\ $$

Question Number 34220 Answers: 0 Comments: 0

calculate I = ∫_0 ^(π/4) cosx ln(tanx)dx .

$${calculate}\:{I}\:=\:\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{4}}} \:{cosx}\:{ln}\left({tanx}\right){dx}\:. \\ $$

Question Number 34219 Answers: 0 Comments: 1

calculate ∫_0 ^(π/4) (dx/(cos^3 x +sin^3 x))

$${calculate}\:\:\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{4}}} \:\:\:\:\frac{{dx}}{{cos}^{\mathrm{3}} {x}\:+{sin}^{\mathrm{3}} {x}} \\ $$

Question Number 34218 Answers: 0 Comments: 0

find ∫(√(tanx))dx 2) calculate ∫_0 ^(π/6) (√(tanx)) dx

$${find}\:\int\sqrt{{tanx}}{dx} \\ $$$$\left.\mathrm{2}\right)\:{calculate}\:\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{6}}} \sqrt{{tanx}}\:{dx} \\ $$

Question Number 34217 Answers: 0 Comments: 1

calculate lim_(n→+∞) n^3 Σ_(k=1) ^n (1/(n^4 +k^2 n^2 +k^4 )) .

$${calculate}\:{lim}_{{n}\rightarrow+\infty} {n}^{\mathrm{3}} \:\sum_{{k}=\mathrm{1}} ^{{n}} \:\:\:\frac{\mathrm{1}}{{n}^{\mathrm{4}} \:+{k}^{\mathrm{2}} {n}^{\mathrm{2}} \:+{k}^{\mathrm{4}} }\:. \\ $$

Question Number 34216 Answers: 0 Comments: 0

let give I =∫_0 ^1 ((ln(x+1))/x)dx and J = ∫_0 ^1 ((ln(1−x))/x)dx 1) prove the existence of I and J 2) calculate I +J and 2I +J 3) find I and J .

$${let}\:{give}\:{I}\:=\int_{\mathrm{0}} ^{\mathrm{1}} \:\frac{{ln}\left({x}+\mathrm{1}\right)}{{x}}{dx}\:{and}\:{J}\:=\:\int_{\mathrm{0}} ^{\mathrm{1}} \:\frac{{ln}\left(\mathrm{1}−{x}\right)}{{x}}{dx} \\ $$$$\left.\mathrm{1}\right)\:{prove}\:{the}\:{existence}\:{of}\:{I}\:{and}\:{J} \\ $$$$\left.\mathrm{2}\right)\:{calculate}\:{I}\:+{J}\:{and}\:\mathrm{2}{I}\:+{J} \\ $$$$\left.\mathrm{3}\right)\:{find}\:{I}\:{and}\:{J}\:. \\ $$

Question Number 34215 Answers: 0 Comments: 0

find the polynome p_n wich verify p_n (0)=0 and ∀ x ∈ R p_n (x)−p_n (x−1) =x^n

$${find}\:{the}\:{polynome}\:{p}_{{n}} \:{wich}\:{verify}\:{p}_{{n}} \left(\mathrm{0}\right)=\mathrm{0}\:{and} \\ $$$$\forall\:{x}\:\in\:{R}\:\:{p}_{{n}} \left({x}\right)−{p}_{{n}} \left({x}−\mathrm{1}\right)\:={x}^{{n}} \\ $$

Question Number 34211 Answers: 1 Comments: 0

let x and y such that 2x^2 +4x−2y=0 y^2 −(x+6)^2 =0 find the possibles value of x+y

$${let}\:{x}\:{and}\:{y}\:{such}\:{that} \\ $$$$\mathrm{2}{x}^{\mathrm{2}} +\mathrm{4}{x}−\mathrm{2}{y}=\mathrm{0} \\ $$$${y}^{\mathrm{2}} −\left({x}+\mathrm{6}\right)^{\mathrm{2}} =\mathrm{0} \\ $$$${find}\:{the}\:{possibles}\:{value}\:{of}\:{x}+{y} \\ $$

Question Number 34208 Answers: 0 Comments: 0

the 15^(th) term of an AP is 34 the sum of the 94^(th) to 100^(th) term is 2000 find the 2^(nd) term and the mean from the sixth term to the 10^(th) term if the 100^(th) term is 360.

$$\:{the}\:\mathrm{15}^{{th}} \:{term}\:{of}\:{an}\:{AP}\:{is}\:\mathrm{34}\:{the}\: \\ $$$${sum}\:{of}\:{the}\:\mathrm{94}^{{th}} \:{to}\:\mathrm{100}^{{th}} \:{term}\:{is} \\ $$$$\mathrm{2000}\:{find}\:{the}\:\mathrm{2}^{{nd}} \:{term}\:{and}\:{the} \\ $$$${mean}\:{from}\:{the}\:{sixth}\:{term}\:{to}\:{the} \\ $$$$\mathrm{10}^{{th}} \:{term}\:{if}\:{the}\:\mathrm{100}^{{th}} \:{term}\:{is} \\ $$$$\:\mathrm{360}. \\ $$

Question Number 34205 Answers: 1 Comments: 1

Question Number 34206 Answers: 0 Comments: 1

Question Number 34202 Answers: 1 Comments: 0

find the gradient of the curve y=(1/x).

$$\:\:{find}\:{the}\:{gradient}\:{of}\:{the}\:{curve} \\ $$$$\:\:\:\:{y}=\frac{\mathrm{1}}{{x}}. \\ $$

Question Number 34196 Answers: 1 Comments: 2

Question Number 34186 Answers: 1 Comments: 1

Let x_1 = 0, x_2 = 1 and x_n = (1/2)(x_(n−1) + x_(n−2) ) Show that x_n = ((2^(n−1) + (−1)^n )/(3 . 2^(n−2) ))

$$\mathrm{Let}\:{x}_{\mathrm{1}} \:=\:\mathrm{0},\:{x}_{\mathrm{2}} \:=\:\mathrm{1}\:\mathrm{and}\:{x}_{{n}} \:=\:\frac{\mathrm{1}}{\mathrm{2}}\left({x}_{{n}−\mathrm{1}} \:+\:{x}_{{n}−\mathrm{2}} \right) \\ $$$$\mathrm{Show}\:\mathrm{that}\: \\ $$$${x}_{{n}} \:=\:\frac{\mathrm{2}^{{n}−\mathrm{1}} \:+\:\left(−\mathrm{1}\right)^{{n}} }{\mathrm{3}\:.\:\mathrm{2}^{{n}−\mathrm{2}} } \\ $$

Question Number 34184 Answers: 1 Comments: 1

if α and β are the roots of the equation 3x^2 + (x/2) − 4= 0 find p is α−β are the roots of x^2 −px + 7 =0

$$\:{if}\:\alpha\:{and}\:\beta\:{are}\:{the}\:{roots}\:{of}\:{the}\:{equation} \\ $$$$\mathrm{3}{x}^{\mathrm{2}} +\:\frac{{x}}{\mathrm{2}}\:−\:\mathrm{4}=\:\mathrm{0}\:{find}\:{p}\:{is}\:\alpha−\beta\:{are} \\ $$$${the}\:{roots}\:{of}\:\:{x}^{\mathrm{2}} −{px}\:+\:\mathrm{7}\:=\mathrm{0} \\ $$

Question Number 34182 Answers: 2 Comments: 0

resolve (x^3 /(x^6 −1)) into partial fraction

$${resolve}\:\frac{{x}^{\mathrm{3}} }{{x}^{\mathrm{6}} −\mathrm{1}}\:{into}\:{partial}\:{fraction} \\ $$

Question Number 34170 Answers: 1 Comments: 0

the distance from A to B is 44km the speed of a car is 2kms^(−1) find the term hence the amount of money needed to buy fuel if for every 6km 10 l of fuel is used which causes 600 box.

$${the}\:{distance}\:{from}\:{A}\:{to}\:{B}\:{is}\:\mathrm{44}{km} \\ $$$${the}\:{speed}\:{of}\:{a}\:{car}\:{is}\:\mathrm{2}{kms}^{−\mathrm{1}} \:{find}\: \\ $$$${the}\:{term}\:{hence}\:{the}\:{amount}\:{of}\: \\ $$$${money}\:{needed}\:{to}\:{buy}\:{fuel}\:{if}\:{for}\: \\ $$$${every}\:\mathrm{6}{km}\:\mathrm{10}\:{l}\:{of}\:{fuel}\:{is}\:{used}\:{which} \\ $$$${causes}\:\mathrm{600}\:{box}. \\ $$

Question Number 34169 Answers: 1 Comments: 0

if the sum S_4 of the first 4 terms of a G P is 24 and the sum S_6 of the first 6 terms is 30 find the first term and common ratio..

$$\:{if}\:{the}\:{sum}\:{S}_{\mathrm{4}} \:{of}\:{the}\:{first}\:\mathrm{4}\:{terms} \\ $$$${of}\:{a}\:{G}\:{P}\:{is}\:\mathrm{24}\:{and}\:{the}\:{sum}\:{S}_{\mathrm{6}} \:{of}\: \\ $$$${the}\:{first}\:\mathrm{6}\:{terms}\:{is}\:\mathrm{30}\:{find}\:{the}\: \\ $$$${first}\:{term}\:{and}\:{common}\:{ratio}.. \\ $$

Question Number 34160 Answers: 1 Comments: 3

prove that lim_(x→0^+ ) ln x∙ln (1+x)=lim_(x→1) ln x∙ln (1+x)

$${prove}\:{that} \\ $$$$\underset{{x}\rightarrow\mathrm{0}^{+} } {\mathrm{lim}ln}\:{x}\centerdot\mathrm{ln}\:\left(\mathrm{1}+{x}\right)=\underset{{x}\rightarrow\mathrm{1}} {\mathrm{lim}ln}\:{x}\centerdot\mathrm{ln}\:\left(\mathrm{1}+{x}\right) \\ $$

Question Number 34159 Answers: 1 Comments: 1

let S=1+2+...+2018 compute S(mod 2)+S(mod8)+S(mod2018)

$${let}\:{S}=\mathrm{1}+\mathrm{2}+...+\mathrm{2018} \\ $$$${compute}\:{S}\left(\mathrm{mod}\:\mathrm{2}\right)+{S}\left(\mathrm{mod8}\right)+{S}\left(\mathrm{mod2018}\right) \\ $$

Question Number 34142 Answers: 0 Comments: 1

what is the derivative of (x+3)(x^2 + 5) and find the n sequence of Σ_(r=n+1 ) ^(2n) (4r^3 −3)

$${what}\:{is}\:{the}\:{derivative}\:{of}\: \\ $$$$\:\:\:\:\:\left(\boldsymbol{{x}}+\mathrm{3}\right)\left(\boldsymbol{{x}}^{\mathrm{2}} +\:\mathrm{5}\right) \\ $$$${and}\:{find}\:{the}\:{n}\:{sequence}\:{of}\: \\ $$$$\:\:\:\underset{{r}={n}+\mathrm{1}\:} {\overset{\mathrm{2}{n}} {\sum}}\:\:\left(\mathrm{4}{r}^{\mathrm{3}} −\mathrm{3}\right) \\ $$

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