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Question Number 72025 Answers: 1 Comments: 1

calculate ∫_0 ^∞ e^(−αx) ln(x)dx with α>0

$${calculate}\:\:\int_{\mathrm{0}} ^{\infty} \:{e}^{−\alpha{x}} {ln}\left({x}\right){dx}\:\:{with}\:\alpha>\mathrm{0} \\ $$

Question Number 72024 Answers: 0 Comments: 1

calculate lim_(n→+∞) ∫_0 ^n (1−(x/n))^n ln(1+x)dx

$${calculate}\:{lim}_{{n}\rightarrow+\infty} \:\:\:\int_{\mathrm{0}} ^{{n}} \left(\mathrm{1}−\frac{{x}}{{n}}\right)^{{n}} {ln}\left(\mathrm{1}+{x}\right){dx} \\ $$

Question Number 72023 Answers: 1 Comments: 1

find lim_(n→+∞) Σ_(1≤i<j≤n) (1/((ij)^2 ))

$${find}\:{lim}_{{n}\rightarrow+\infty} \:\:\:\sum_{\mathrm{1}\leqslant{i}<{j}\leqslant{n}} \:\:\frac{\mathrm{1}}{\left({ij}\right)^{\mathrm{2}} } \\ $$

Question Number 72021 Answers: 0 Comments: 2

calculate interms of n U_n =Σ_(1≤i<j≤n) sin(((iπ)/n))sin(((jπ)/n))

$${calculate}\:{interms}\:{of}\:{n}\:\:{U}_{{n}} =\sum_{\mathrm{1}\leqslant{i}<{j}\leqslant{n}} \:{sin}\left(\frac{{i}\pi}{{n}}\right){sin}\left(\frac{{j}\pi}{{n}}\right) \\ $$

Question Number 72020 Answers: 1 Comments: 1

calculate Σ_(k=0) ^n C_n ^k cos(((kπ)/(2n)))

$${calculate}\:\sum_{{k}=\mathrm{0}} ^{{n}} \:{C}_{{n}} ^{{k}} \:{cos}\left(\frac{{k}\pi}{\mathrm{2}{n}}\right) \\ $$

Question Number 72018 Answers: 0 Comments: 1

calculate Σ_(n=1) ^∞ (((−1)^n )/(n^3 (n+1)^2 ))

$${calculate}\:\sum_{{n}=\mathrm{1}} ^{\infty} \:\:\:\frac{\left(−\mathrm{1}\right)^{{n}} }{{n}^{\mathrm{3}} \left({n}+\mathrm{1}\right)^{\mathrm{2}} } \\ $$

Question Number 72017 Answers: 0 Comments: 0

find ∫ ((x+(√(4+x^2 )))/(x−(√(4+3x^2 ))))dx

$${find}\:\int\:\:\frac{{x}+\sqrt{\mathrm{4}+{x}^{\mathrm{2}} }}{{x}−\sqrt{\mathrm{4}+\mathrm{3}{x}^{\mathrm{2}} }}{dx} \\ $$

Question Number 72016 Answers: 1 Comments: 2

find U_n =∫_0 ^∞ ((cos(nx))/((3+nx^2 )^2 ))dx with n integr

$${find}\:{U}_{{n}} =\int_{\mathrm{0}} ^{\infty} \:\:\:\frac{{cos}\left({nx}\right)}{\left(\mathrm{3}+{nx}^{\mathrm{2}} \right)^{\mathrm{2}} }{dx}\:\:{with}\:{n}\:{integr} \\ $$

Question Number 72015 Answers: 0 Comments: 1

find ∫ ((arctan(x−(1/x)))/(x^2 +1))dx

$${find}\:\int\:\:\frac{{arctan}\left({x}−\frac{\mathrm{1}}{{x}}\right)}{{x}^{\mathrm{2}} \:+\mathrm{1}}{dx} \\ $$

Question Number 72012 Answers: 0 Comments: 1

find f(α) =∫_0 ^∞ ((arctan(αx))/((x^2 +3)^2 ))dx with α real.

$${find}\:{f}\left(\alpha\right)\:=\int_{\mathrm{0}} ^{\infty} \:\:\frac{{arctan}\left(\alpha{x}\right)}{\left({x}^{\mathrm{2}} +\mathrm{3}\right)^{\mathrm{2}} }{dx}\:\:{with}\:\alpha\:{real}. \\ $$

Question Number 72011 Answers: 1 Comments: 1

calculate ∫_0 ^∞ ((ln(3+x^2 ))/((x^2 +1)^2 ))dx

$${calculate}\:\int_{\mathrm{0}} ^{\infty} \:\:\frac{{ln}\left(\mathrm{3}+{x}^{\mathrm{2}} \right)}{\left({x}^{\mathrm{2}} +\mathrm{1}\right)^{\mathrm{2}} }{dx} \\ $$

Question Number 71993 Answers: 1 Comments: 0

Question Number 71986 Answers: 1 Comments: 0

Question Number 71984 Answers: 1 Comments: 0

x, y ∈ I ⊂ R prove that ∣cosx − siny∣ ≤ ∣x − y∣

$$\:{x},\:{y}\:\in\:{I}\:\subset\:\mathbb{R}\:\:{prove}\:\:{that}\:\:\mid{cosx}\:−\:{siny}\mid\:\leqslant\:\mid{x}\:−\:{y}\mid \\ $$

Question Number 71983 Answers: 1 Comments: 0

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Question Number 74587 Answers: 0 Comments: 0

If sum of n arithmetic means between two number is 20.if last mean is double of 1st mean and one is three times of another number. find the numbers.

$$\mathrm{If}\:\mathrm{sum}\:\mathrm{of}\:\mathrm{n}\:\mathrm{arithmetic}\:\mathrm{means}\:\mathrm{between}\: \\ $$$$\mathrm{two}\:\mathrm{number}\:\mathrm{is}\:\mathrm{20}.\mathrm{if}\:\mathrm{last}\:\mathrm{mean}\:\mathrm{is}\:\mathrm{double} \\ $$$$\mathrm{of}\:\mathrm{1st}\:\mathrm{mean}\:\mathrm{and}\:\mathrm{one}\:\mathrm{is}\:\mathrm{three}\:\mathrm{times}\:\mathrm{of} \\ $$$$\mathrm{another}\:\mathrm{number}.\:\mathrm{find}\:\mathrm{the}\:\mathrm{numbers}. \\ $$$$ \\ $$

Question Number 71972 Answers: 0 Comments: 2

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Question Number 71964 Answers: 5 Comments: 1

Question Number 71960 Answers: 1 Comments: 0

Question Number 71957 Answers: 0 Comments: 0

Question Number 71955 Answers: 1 Comments: 1

Question Number 71946 Answers: 1 Comments: 0

Question Number 71939 Answers: 1 Comments: 1

Question Number 71944 Answers: 1 Comments: 0

Question Number 71925 Answers: 1 Comments: 0

Solve (√(1 − x^2 )) = x^(4/3)

$$\:\boldsymbol{{Solve}}\:\:\sqrt{\mathrm{1}\:−\:\boldsymbol{{x}}^{\mathrm{2}} }\:=\:\boldsymbol{{x}}^{\mathrm{4}/\mathrm{3}} \\ $$

Question Number 71921 Answers: 0 Comments: 0

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