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Question Number 142935 Answers: 2 Comments: 0

Question Number 142931 Answers: 0 Comments: 0

Question Number 142922 Answers: 2 Comments: 0

lim_(x→1) ((sin(x+1))/(2x−(√(x^2 +3))))=?

$$\underset{{x}\rightarrow\mathrm{1}} {\mathrm{lim}}\frac{\mathrm{sin}\left({x}+\mathrm{1}\right)}{\mathrm{2}{x}−\sqrt{{x}^{\mathrm{2}} +\mathrm{3}}}=? \\ $$

Question Number 142927 Answers: 2 Comments: 0

Question Number 142920 Answers: 1 Comments: 2

A student did not notice that the multiplication sign between two 7−digits numbers amd wrote one 14−digits number which turned out to be 3 times the would be product. What are the initial numbers ?

$$\mathrm{A}\:\mathrm{student}\:\mathrm{did}\:\mathrm{not}\:\mathrm{notice}\:\mathrm{that}\:\mathrm{the}\:\mathrm{multiplication} \\ $$$$\mathrm{sign}\:\mathrm{between}\:\mathrm{two}\:\mathrm{7}−\mathrm{digits}\:\mathrm{numbers}\:\mathrm{amd}\:\mathrm{wrote} \\ $$$$\mathrm{one}\:\mathrm{14}−\mathrm{digits}\:\mathrm{number}\:\mathrm{which}\:\mathrm{turned}\:\mathrm{out}\:\mathrm{to}\:\mathrm{be} \\ $$$$\mathrm{3}\:\mathrm{times}\:\mathrm{the}\:\mathrm{would}\:\mathrm{be}\:\mathrm{product}.\:\mathrm{What}\:\mathrm{are}\:\mathrm{the}\:\mathrm{initial} \\ $$$$\mathrm{numbers}\:? \\ $$

Question Number 142917 Answers: 1 Comments: 0

∫(sin^7 (x))dx

$$\int\left(\boldsymbol{\mathrm{sin}}^{\mathrm{7}} \left(\boldsymbol{\mathrm{x}}\right)\right)\boldsymbol{\mathrm{dx}} \\ $$

Question Number 142915 Answers: 0 Comments: 1

Question Number 142914 Answers: 0 Comments: 0

Let a≥b≥c≥d>0 and a+b+c+d = 4. Prove that (((√a)+(√b)+(√c))/3) ≤ (1/( (√d))) Prove if ∀n∈N^+ , then (((a)^(1/n) +(b)^(1/n) +(c)^(1/n) )/3) ≤ (1/( (d)^(1/n) ))

$$\mathrm{Let}\:{a}\geqslant{b}\geqslant{c}\geqslant{d}>\mathrm{0}\:\mathrm{and}\:{a}+{b}+{c}+{d}\:=\:\mathrm{4}. \\ $$$$\mathrm{Prove}\:\mathrm{that} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\frac{\sqrt{{a}}+\sqrt{{b}}+\sqrt{{c}}}{\mathrm{3}}\:\leqslant\:\frac{\mathrm{1}}{\:\sqrt{{d}}} \\ $$$$\mathrm{Prove}\:\mathrm{if}\:\forall{n}\in\mathbb{N}^{+} ,\:\mathrm{then} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\frac{\sqrt[{{n}}]{{a}}+\sqrt[{{n}}]{{b}}+\sqrt[{{n}}]{{c}}}{\mathrm{3}}\:\leqslant\:\frac{\mathrm{1}}{\:\sqrt[{{n}}]{{d}}} \\ $$$$ \\ $$

Question Number 142912 Answers: 0 Comments: 0

Question Number 142910 Answers: 2 Comments: 0

Question Number 142906 Answers: 1 Comments: 0

If abc=1 and a,b,c>0 prove that (a/(b^2 (c+1)))+(b/(c^2 (a+1)))+(c/(a^2 (b+1))) ≥ (3/2)

$${If}\:{abc}=\mathrm{1}\:{and}\:{a},{b},{c}>\mathrm{0}\:{prove} \\ $$$${that}\:\frac{{a}}{{b}^{\mathrm{2}} \left({c}+\mathrm{1}\right)}+\frac{{b}}{{c}^{\mathrm{2}} \left({a}+\mathrm{1}\right)}+\frac{{c}}{{a}^{\mathrm{2}} \left({b}+\mathrm{1}\right)}\:\geqslant\:\frac{\mathrm{3}}{\mathrm{2}} \\ $$

Question Number 142904 Answers: 1 Comments: 0

If 2+log _2 (x)=3+log _3 (y)=log _6 (x−4y) then the value of (1/(2y))−(2/x)=?

$${If}\:\mathrm{2}+\mathrm{log}\:_{\mathrm{2}} \left({x}\right)=\mathrm{3}+\mathrm{log}\:_{\mathrm{3}} \left({y}\right)=\mathrm{log}\:_{\mathrm{6}} \left({x}−\mathrm{4}{y}\right) \\ $$$${then}\:{the}\:{value}\:{of}\:\frac{\mathrm{1}}{\mathrm{2}{y}}−\frac{\mathrm{2}}{{x}}=? \\ $$

Question Number 142902 Answers: 1 Comments: 0

I_n =∫_0 ^( _ (π/2)) (sin x)^n dx with integration by parts, prove that : I_(n+2) = ((n+1)/(n+2)) . I_n

$$\mathrm{I}_{{n}} =\int_{\mathrm{0}} ^{\:_{} \frac{\pi}{\mathrm{2}}} \:\left(\mathrm{sin}\:{x}\right)^{{n}} \:{dx} \\ $$$$\boldsymbol{\mathrm{with}}\:\boldsymbol{\mathrm{integration}}\:\boldsymbol{\mathrm{by}}\:\boldsymbol{\mathrm{parts}},\:\boldsymbol{\mathrm{prove}}\:\boldsymbol{\mathrm{that}}\::\: \\ $$$$\mathrm{I}_{{n}+\mathrm{2}} \:=\:\frac{{n}+\mathrm{1}}{{n}+\mathrm{2}}\:.\:\mathrm{I}_{{n}} \\ $$

Question Number 142901 Answers: 0 Comments: 2

Question Number 142900 Answers: 0 Comments: 3

Question Number 142898 Answers: 0 Comments: 0

Question Number 142903 Answers: 0 Comments: 0

Question Number 142893 Answers: 1 Comments: 0

.....mathematical .....analysis...... f ∈ C [0,1] and ∫_0 ^( 1) x^n f(x)dx=(1/(n+2)) , n∈N prove f(x):=x .....

$$\:\:\:\:\:\:\:\:\: \\ $$$$\:\:\:\:\:\:\:\:\:\:\:.....{mathematical}\:.....{analysis}...... \\ $$$$\:\:\:\:\:\:\:{f}\:\in\:{C}\:\left[\mathrm{0},\mathrm{1}\right]\:{and}\:\:\int_{\mathrm{0}} ^{\:\mathrm{1}} {x}^{{n}} {f}\left({x}\right){dx}=\frac{\mathrm{1}}{{n}+\mathrm{2}}\:,\:{n}\in\mathbb{N} \\ $$$$\:\:\:\:\:\:\:\:{prove}\:\:{f}\left({x}\right):={x}\:..... \\ $$

Question Number 142886 Answers: 2 Comments: 0

Evaluate ∫_0 ^( (1/2)) ((4x^2 )/( (√(1−x^2 )) )) dx

$$\mathrm{Evaluate}\:\int_{\mathrm{0}} ^{\:\frac{\mathrm{1}}{\mathrm{2}}} \frac{\mathrm{4}{x}^{\mathrm{2}} }{\:\sqrt{\mathrm{1}−{x}^{\mathrm{2}} }\:}\:{dx} \\ $$

Question Number 142882 Answers: 2 Comments: 1

Question Number 142880 Answers: 1 Comments: 0

Prove that 𝛗(n)=nΠ_k (1−(1/p_k )) φ(n):Euler totient function

$$\:{Prove}\:{that}\:\boldsymbol{\phi}\left({n}\right)={n}\underset{{k}} {\prod}\left(\mathrm{1}−\frac{\mathrm{1}}{{p}_{{k}} }\right)\:\:\phi\left({n}\right):{Euler}\:{totient}\:{function} \\ $$

Question Number 142875 Answers: 1 Comments: 0

Prove that ζ(s)=Π_(prime) (1/(1−p^(−s) ))

$${Prove}\:{that}\:\zeta\left({s}\right)=\underset{{prime}} {\prod}\:\frac{\mathrm{1}}{\mathrm{1}−{p}^{−{s}} } \\ $$

Question Number 142871 Answers: 0 Comments: 0

determine arctan(x+iy) at form u(x,y)+iv(x,y)

$$\mathrm{determine}\:\mathrm{arctan}\left(\mathrm{x}+\mathrm{iy}\right)\:\mathrm{at}\:\mathrm{form}\:\mathrm{u}\left(\mathrm{x},\mathrm{y}\right)+\mathrm{iv}\left(\mathrm{x},\mathrm{y}\right) \\ $$

Question Number 142870 Answers: 1 Comments: 0

Question Number 142869 Answers: 2 Comments: 0

calculate ∫_(−∞) ^(+∞) ((x^2 dx)/((x^2 −x+3)^2 ))

$$\mathrm{calculate}\:\int_{−\infty} ^{+\infty} \:\:\frac{\mathrm{x}^{\mathrm{2}} \mathrm{dx}}{\left(\mathrm{x}^{\mathrm{2}} −\mathrm{x}+\mathrm{3}\right)^{\mathrm{2}} } \\ $$

Question Number 142863 Answers: 1 Comments: 0

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