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Question Number 142414    Answers: 1   Comments: 2

4sin^2 18° + 2sin18° + 2,5 = ?

$$\mathrm{4}{sin}^{\mathrm{2}} \mathrm{18}°\:+\:\mathrm{2}{sin}\mathrm{18}°\:+\:\mathrm{2},\mathrm{5}\:=\:? \\ $$

Question Number 142408    Answers: 1   Comments: 1

Question Number 142404    Answers: 2   Comments: 1

Question Number 142401    Answers: 1   Comments: 1

Question Number 142395    Answers: 0   Comments: 0

Question Number 142397    Answers: 0   Comments: 0

Question Number 142393    Answers: 1   Comments: 0

Σ_(k=0) ^(n−1) sec^2 (((kπ)/n))=n^2 ......???

$$\underset{\mathrm{k}=\mathrm{0}} {\overset{\mathrm{n}−\mathrm{1}} {\sum}}\mathrm{sec}^{\mathrm{2}} \left(\frac{\mathrm{k}\pi}{\mathrm{n}}\right)=\mathrm{n}^{\mathrm{2}} ......??? \\ $$

Question Number 142389    Answers: 1   Comments: 0

∫(e^x /(cosx))dx

$$\int\frac{{e}^{{x}} }{{cosx}}{dx} \\ $$

Question Number 142388    Answers: 1   Comments: 0

n ∈ N, b, a ∈ N ; a≠0. In base 10; n=aabb^(−) 1. show that n is not prime. 2. Give conditions on b such that n is perfect square. 3. Determinate n such that n is a perfect square.

$$\mathrm{n}\:\in\:\mathbb{N},\:\mathrm{b},\:\mathrm{a}\:\in\:\mathbb{N}\:;\:\mathrm{a}\neq\mathrm{0}. \\ $$$$\mathrm{In}\:\mathrm{base}\:\mathrm{10};\:\mathrm{n}=\overline {\mathrm{aabb}}\: \\ $$$$\mathrm{1}.\:\mathrm{show}\:\mathrm{that}\:\mathrm{n}\:\mathrm{is}\:\mathrm{not}\:\mathrm{prime}. \\ $$$$\mathrm{2}.\:\mathrm{Give}\:\mathrm{conditions}\:\mathrm{on}\:\mathrm{b}\:\mathrm{such}\:\mathrm{that} \\ $$$$\mathrm{n}\:\mathrm{is}\:\mathrm{perfect}\:\mathrm{square}. \\ $$$$\mathrm{3}.\:\mathrm{Determinate}\:\mathrm{n}\:\mathrm{such}\:\mathrm{that}\:\mathrm{n}\:\mathrm{is}\:\mathrm{a}\: \\ $$$$\mathrm{perfect}\:\mathrm{square}. \\ $$

Question Number 142383    Answers: 1   Comments: 1

Question Number 142379    Answers: 0   Comments: 1

Show that 1+3n<n^2 for every positive integer n≥4

$${Show}\:{that}\:\mathrm{1}+\mathrm{3}{n}<{n}^{\mathrm{2}} \:{for}\:{every}\:{positive}\:{integer}\:{n}\geqslant\mathrm{4} \\ $$

Question Number 142365    Answers: 1   Comments: 0

calculate ∫ (√(1+e^x +e^(2x) ))dx

$$\mathrm{calculate}\:\int\:\:\sqrt{\mathrm{1}+\mathrm{e}^{\mathrm{x}} \:+\mathrm{e}^{\mathrm{2x}} }\mathrm{dx} \\ $$

Question Number 142362    Answers: 1   Comments: 0

prove that: ∫_0 ^( ∞) ln((1/x)).j_0 (x)dx:= γ+ln(2) Hint:(1) j_0 (x)=Σ_(n=0) ^∞ (((−1)^n x^(2n) )/(2^(2n) .Γ^2 (n+1))) (Bessel function) Hint:2 L [ j_0 (x)]=(1/( (√(1+s^2 ))))

$$ \\ $$$$\:\:\:\:\:\:\:\:\:{prove}\:\:{that}: \\ $$$$\:\:\:\:\:\:\int_{\mathrm{0}} ^{\:\infty} {ln}\left(\frac{\mathrm{1}}{{x}}\right).{j}_{\mathrm{0}} \left({x}\right){dx}:=\:\gamma+{ln}\left(\mathrm{2}\right)\: \\ $$$$\:\:\:\:\:{Hint}:\left(\mathrm{1}\right) \\ $$$$\:\:\:\:\:\:\:{j}_{\mathrm{0}} \left({x}\right)=\underset{{n}=\mathrm{0}} {\overset{\infty} {\sum}}\frac{\left(−\mathrm{1}\right)^{{n}} {x}^{\mathrm{2}{n}} }{\mathrm{2}^{\mathrm{2}{n}} .\Gamma^{\mathrm{2}} \left({n}+\mathrm{1}\right)}\:\left({Bessel}\:{function}\right) \\ $$$$\:\:\:\:{Hint}:\mathrm{2}\: \\ $$$$\:\:\:\:\:\:\:\:\:\:\mathscr{L}\:\left[\:{j}_{\mathrm{0}} \left({x}\right)\right]=\frac{\mathrm{1}}{\:\sqrt{\mathrm{1}+{s}^{\mathrm{2}} }} \\ $$

Question Number 142361    Answers: 1   Comments: 0

Question Number 142359    Answers: 0   Comments: 0

Question Number 142358    Answers: 1   Comments: 0

∫_0 ^∞ x^(n−1) log_e (1−x)dx

$$\int_{\mathrm{0}} ^{\infty} {x}^{{n}−\mathrm{1}} {log}_{{e}} \left(\mathrm{1}−{x}\right){dx} \\ $$

Question Number 142356    Answers: 0   Comments: 0

Question Number 142351    Answers: 1   Comments: 0

lim_(x→0) ((((27+x))^(1/(3 )) −((27−x))^(1/(3 )) )/( (x^2 )^(1/(3 )) + (x^3 )^(1/(4 )) )) =?

$$\:\:\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\frac{\sqrt[{\mathrm{3}\:}]{\mathrm{27}+{x}}−\sqrt[{\mathrm{3}\:}]{\mathrm{27}−{x}}}{\:\sqrt[{\mathrm{3}\:}]{{x}^{\mathrm{2}} }\:+\:\sqrt[{\mathrm{4}\:}]{{x}^{\mathrm{3}} }}\:=? \\ $$

Question Number 142347    Answers: 1   Comments: 1

calculate ∫_0 ^∞ ((x^2 logx)/(x^6 +1))dx

$$\mathrm{calculate}\:\int_{\mathrm{0}} ^{\infty} \:\frac{\mathrm{x}^{\mathrm{2}} \mathrm{logx}}{\mathrm{x}^{\mathrm{6}} \:+\mathrm{1}}\mathrm{dx} \\ $$

Question Number 142346    Answers: 0   Comments: 0

calculate ∫_0 ^∞ ((log(1+x^3 ))/(1+x^4 ))dx

$$\mathrm{calculate}\:\int_{\mathrm{0}} ^{\infty} \:\:\frac{\mathrm{log}\left(\mathrm{1}+\mathrm{x}^{\mathrm{3}} \right)}{\mathrm{1}+\mathrm{x}^{\mathrm{4}} }\mathrm{dx} \\ $$

Question Number 142344    Answers: 0   Comments: 3

𝛗:=∫^( e) _(1/e) {(1/(ln(x)))+ln(ln(x))}dx

$$ \\ $$$$\:\:\:\:\:\:\boldsymbol{\phi}:=\underset{\frac{\mathrm{1}}{{e}}} {\int}^{\:{e}} \left\{\frac{\mathrm{1}}{{ln}\left({x}\right)}+{ln}\left({ln}\left({x}\right)\right)\right\}{dx} \\ $$

Question Number 142341    Answers: 1   Comments: 1

Question Number 142338    Answers: 2   Comments: 0

∫_0 ^∞ ((sinx)/x^μ )dx =?

$$\int_{\mathrm{0}} ^{\infty} \frac{{sinx}}{{x}^{\mu} }{dx}\:\:=?\:\:\: \\ $$

Question Number 142325    Answers: 0   Comments: 0

Given that a ≥ 1 ≥ b > 0. Prove the followings: (1) (1/2)(a−b)^2 ≤ (a−1)^2 +(1−b)^2 ≤ (a−b)^2 (2) (1/4)(a−b)^3 ≤ (a−1)^3 +(1−b)^3 ≤ (a−b)^3

$$\mathrm{Given}\:\mathrm{that}\:{a}\:\geqslant\:\mathrm{1}\:\geqslant\:{b}\:>\:\mathrm{0}.\:\mathrm{Prove}\:\mathrm{the}\:\mathrm{followings}:\:\:\:\:\:\:\:\:\:\:\: \\ $$$$\left(\mathrm{1}\right)\:\:\:\:\:\:\:\:\:\:\:\:\:\:\frac{\mathrm{1}}{\mathrm{2}}\left({a}−{b}\right)^{\mathrm{2}} \:\leqslant\:\left({a}−\mathrm{1}\right)^{\mathrm{2}} +\left(\mathrm{1}−{b}\right)^{\mathrm{2}} \:\leqslant\:\left({a}−{b}\right)^{\mathrm{2}} \:\:\:\: \\ $$$$\left(\mathrm{2}\right)\:\:\:\:\:\:\:\:\:\:\:\:\:\:\frac{\mathrm{1}}{\mathrm{4}}\left({a}−{b}\right)^{\mathrm{3}} \:\leqslant\:\left({a}−\mathrm{1}\right)^{\mathrm{3}} +\left(\mathrm{1}−{b}\right)^{\mathrm{3}} \:\leqslant\:\left({a}−{b}\right)^{\mathrm{3}} \\ $$$$ \\ $$

Question Number 142349    Answers: 2   Comments: 0

Σ_(n=0) ^∞ ((ζ(2n+2)(−1)^n )/4^n )=?

$$ \\ $$$$\:\:\:\:\:\:\:\:\underset{{n}=\mathrm{0}} {\overset{\infty} {\sum}}\frac{\zeta\left(\mathrm{2}{n}+\mathrm{2}\right)\left(−\mathrm{1}\right)^{{n}} }{\mathrm{4}^{{n}} }=? \\ $$

Question Number 142348    Answers: 1   Comments: 0

(((4−(√(15))))^(1/6) /( (√(4−(√(15)))) ∙ ((4+(√(15))))^(1/3) )) = ?

$$\frac{\sqrt[{\mathrm{6}}]{\mathrm{4}−\sqrt{\mathrm{15}}}}{\:\sqrt{\mathrm{4}−\sqrt{\mathrm{15}}}\:\centerdot\:\sqrt[{\mathrm{3}}]{\mathrm{4}+\sqrt{\mathrm{15}}}}\:=\:? \\ $$

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