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Question Number 96602 Answers: 3 Comments: 0

lim_(n→+∞) (1/(√n)) Σ_(k=1) ^n (√k)

$$\underset{\mathrm{n}\rightarrow+\infty} {\mathrm{lim}}\frac{\mathrm{1}}{\sqrt{\mathrm{n}}}\:\underset{\mathrm{k}=\mathrm{1}} {\overset{\mathrm{n}} {\sum}}\sqrt{\mathrm{k}} \\ $$

Question Number 96593 Answers: 2 Comments: 0

let f(x) =arctan(x^n ) with n integr natural 1) calculate f^′ (x) and f^((2)) (x) 2) calculate f^((n)) (x) and f^((n)) (0) 3)developp f at integr serie

$$\mathrm{let}\:\mathrm{f}\left(\mathrm{x}\right)\:=\mathrm{arctan}\left(\mathrm{x}^{\mathrm{n}} \right)\:\mathrm{with}\:\mathrm{n}\:\mathrm{integr}\:\mathrm{natural} \\ $$$$\left.\mathrm{1}\right)\:\mathrm{calculate}\:\mathrm{f}^{'} \left(\mathrm{x}\right)\:\mathrm{and}\:\mathrm{f}^{\left(\mathrm{2}\right)} \left(\mathrm{x}\right) \\ $$$$\left.\mathrm{2}\right)\:\mathrm{calculate}\:\mathrm{f}^{\left(\mathrm{n}\right)} \left(\mathrm{x}\right)\:\mathrm{and}\:\mathrm{f}^{\left(\mathrm{n}\right)} \left(\mathrm{0}\right) \\ $$$$\left.\mathrm{3}\right)\mathrm{developp}\:\mathrm{f}\:\mathrm{at}\:\mathrm{integr}\:\mathrm{serie} \\ $$

Question Number 96596 Answers: 2 Comments: 0

x(x+1) (dy/dx)−(2x+1)y = 0

$${x}\left({x}+\mathrm{1}\right)\:\frac{{dy}}{{dx}}−\left(\mathrm{2}{x}+\mathrm{1}\right){y}\:=\:\mathrm{0} \\ $$

Question Number 96595 Answers: 0 Comments: 0

∫_0 ^1 x^(4035) (x^4 +1)^(2017) (3x+1)^4 dx

$$\int_{\mathrm{0}} ^{\mathrm{1}} {x}^{\mathrm{4035}} \left({x}^{\mathrm{4}} +\mathrm{1}\right)^{\mathrm{2017}} \left(\mathrm{3}{x}+\mathrm{1}\right)^{\mathrm{4}} {dx} \\ $$

Question Number 96586 Answers: 1 Comments: 0

find the minimum value of f(x)=x^x for x∈R^+

$${find}\:{the}\:{minimum}\:{value}\:{of}\: \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:{f}\left({x}\right)={x}^{{x}} \\ $$$${for}\:{x}\in\mathbb{R}^{+} \\ $$

Question Number 96584 Answers: 1 Comments: 4

Question Number 96581 Answers: 1 Comments: 0

Question Number 96600 Answers: 1 Comments: 0

Question Number 96571 Answers: 2 Comments: 0

solve y^(′′) −2y^′ +y =(x+1)^2 e^(−x) with y(0)=−1 ,y^′ (0)=0 ,y^((2)) (0) =1

$$\mathrm{solve}\:\mathrm{y}^{''} −\mathrm{2y}^{'} \:+\mathrm{y}\:=\left(\mathrm{x}+\mathrm{1}\right)^{\mathrm{2}} \:\mathrm{e}^{−\mathrm{x}} \:\mathrm{with}\:\mathrm{y}\left(\mathrm{0}\right)=−\mathrm{1}\:,\mathrm{y}^{'} \left(\mathrm{0}\right)=\mathrm{0}\:,\mathrm{y}^{\left(\mathrm{2}\right)} \left(\mathrm{0}\right)\:=\mathrm{1} \\ $$

Question Number 96570 Answers: 0 Comments: 0

solve xy^(′′) −(x^3 +1)y^′ +2y =x^2 e^(−x) with y(o)=1 and y^′ (0)=−2

$$\mathrm{solve}\:\:\mathrm{xy}^{''} \:−\left(\mathrm{x}^{\mathrm{3}} +\mathrm{1}\right)\mathrm{y}^{'} \:+\mathrm{2y}\:=\mathrm{x}^{\mathrm{2}} \:\mathrm{e}^{−\mathrm{x}} \:\mathrm{with}\:\:\mathrm{y}\left(\mathrm{o}\right)=\mathrm{1}\:\mathrm{and}\:\mathrm{y}^{'} \left(\mathrm{0}\right)=−\mathrm{2} \\ $$

Question Number 96567 Answers: 1 Comments: 1

Question Number 96558 Answers: 1 Comments: 0

Let m and n be two positive integers satisfy (m/n) = (1/(10×12))+(1/(12×14))+(1/(14×16))+...+(1/(2012×2014)) find the smallest possible value of m+n

$$\mathrm{Let}\:\mathrm{m}\:\mathrm{and}\:\mathrm{n}\:\mathrm{be}\:\mathrm{two}\:\mathrm{positive}\:\mathrm{integers}\: \\ $$$$\mathrm{satisfy}\:\frac{\mathrm{m}}{\mathrm{n}}\:=\:\frac{\mathrm{1}}{\mathrm{10}×\mathrm{12}}+\frac{\mathrm{1}}{\mathrm{12}×\mathrm{14}}+\frac{\mathrm{1}}{\mathrm{14}×\mathrm{16}}+...+\frac{\mathrm{1}}{\mathrm{2012}×\mathrm{2014}} \\ $$$$\mathrm{find}\:\mathrm{the}\:\mathrm{smallest}\:\mathrm{possible}\:\mathrm{value}\:\mathrm{of} \\ $$$$\mathrm{m}+\mathrm{n}\: \\ $$

Question Number 96554 Answers: 1 Comments: 2

∫(dx/(x!))=?

$$\int\frac{\mathrm{dx}}{\mathrm{x}!}=? \\ $$

Question Number 96548 Answers: 0 Comments: 1

Question Number 96543 Answers: 1 Comments: 0

Question Number 96542 Answers: 0 Comments: 6

Question Number 96538 Answers: 1 Comments: 0

Question Number 96530 Answers: 0 Comments: 1

If: (1+m)^3 −3m^3 =2 0≤m≤1 Find: (1+m)^2 −3m^2

$$\boldsymbol{\mathrm{If}}:\:\left(\mathrm{1}+{m}\right)^{\mathrm{3}} −\mathrm{3}{m}^{\mathrm{3}} =\mathrm{2}\:\:\:\:\:\:\:\:\mathrm{0}\leqslant{m}\leqslant\mathrm{1} \\ $$$$\boldsymbol{\mathrm{Find}}:\:\left(\mathrm{1}+{m}\right)^{\mathrm{2}} −\mathrm{3}{m}^{\mathrm{2}} \\ $$

Question Number 96527 Answers: 2 Comments: 1

proof that 1^2 +2^2 +3^2 +....+n^2 =((n(2n+1)(n+1))/6)

$$\mathrm{proof}\:\mathrm{that}\:\mathrm{1}^{\mathrm{2}} +\mathrm{2}^{\mathrm{2}} +\mathrm{3}^{\mathrm{2}} +....+\mathrm{n}^{\mathrm{2}} =\frac{\mathrm{n}\left(\mathrm{2n}+\mathrm{1}\right)\left(\mathrm{n}+\mathrm{1}\right)}{\mathrm{6}} \\ $$

Question Number 96524 Answers: 0 Comments: 1

∫(sinx^(cosx^(sinx) ) −cosx^(sinx^(cosx) ) )dx

$$\int\left(\mathrm{sinx}^{\mathrm{cosx}^{\mathrm{sinx}} } −\mathrm{cosx}^{\mathrm{sinx}^{\mathrm{cosx}} } \right)\mathrm{dx} \\ $$

Question Number 96523 Answers: 0 Comments: 0

Question Number 96514 Answers: 1 Comments: 3

∫_0 ^1 (1/(1+[(1/x)]))dx=?

$$\underset{\mathrm{0}} {\overset{\mathrm{1}} {\int}}\frac{\mathrm{1}}{\mathrm{1}+\left[\frac{\mathrm{1}}{\boldsymbol{{x}}}\right]}\boldsymbol{{dx}}=? \\ $$

Question Number 96508 Answers: 1 Comments: 0

Question Number 96505 Answers: 2 Comments: 2

((√((8)^(1/(4 )) −(√((√2)+1))))/((√((8)^(1/(4 )) +(√((√2)−1))))−(√((8)^(1/(4 )) −(√((√2)−1)))))) ?

$$\frac{\sqrt{\sqrt[{\mathrm{4}\:\:}]{\mathrm{8}}−\sqrt{\sqrt{\mathrm{2}}+\mathrm{1}}}}{\sqrt{\sqrt[{\mathrm{4}\:\:}]{\mathrm{8}}+\sqrt{\sqrt{\mathrm{2}}−\mathrm{1}}}−\sqrt{\sqrt[{\mathrm{4}\:\:}]{\mathrm{8}}−\sqrt{\sqrt{\mathrm{2}}−\mathrm{1}}}}\:? \\ $$

Question Number 96500 Answers: 2 Comments: 0

If x and y real number satisfy (x+5)^2 +(y−12)^2 =196 , then the minimum value of x^2 +y^2 is

$$\mathrm{If}\:{x}\:{and}\:{y}\:{real}\:{number}\:{satisfy} \\ $$$$\left({x}+\mathrm{5}\right)^{\mathrm{2}} +\left(\mathrm{y}−\mathrm{12}\right)^{\mathrm{2}} =\mathrm{196}\:,\:\mathrm{then}\: \\ $$$$\mathrm{the}\:\mathrm{minimum}\:\mathrm{value}\:\mathrm{of}\:{x}^{\mathrm{2}} +{y}^{\mathrm{2}} \:{is}\: \\ $$

Question Number 96495 Answers: 2 Comments: 0

calculateI = ∫_0 ^(π/2) ln(cosx +sinx)dx and J =∫_0 ^(π/2) ln(cosx−sinx)dx

$$\mathrm{calculateI}\:=\:\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \mathrm{ln}\left(\mathrm{cosx}\:+\mathrm{sinx}\right)\mathrm{dx} \\ $$$$\mathrm{and}\:\mathrm{J}\:=\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \mathrm{ln}\left(\mathrm{cosx}−\mathrm{sinx}\right)\mathrm{dx} \\ $$

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