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Question Number 55776 Answers: 1 Comments: 0

lim_(n→∞) Σ_(r=1) ^n ((r^3 /(r^4 +n^4 ))) =

$$\underset{{n}\rightarrow\infty} {\mathrm{lim}}\:\:\underset{{r}=\mathrm{1}} {\overset{{n}} {\sum}}\:\left(\frac{{r}^{\mathrm{3}} }{{r}^{\mathrm{4}} +{n}^{\mathrm{4}} }\right)\:= \\ $$

Question Number 55775 Answers: 1 Comments: 1

∫_( 0) ^1 ∣sin 2π x∣ dx =

$$\:\underset{\:\mathrm{0}} {\overset{\mathrm{1}} {\int}}\:\:\mid\mathrm{sin}\:\mathrm{2}\pi\:{x}\mid\:{dx}\:= \\ $$

Question Number 55774 Answers: 1 Comments: 0

∫_(−1/2) ^(1/2) cos x log (((1+x)/(1−x))) dx =

$$\:\underset{−\mathrm{1}/\mathrm{2}} {\overset{\mathrm{1}/\mathrm{2}} {\int}}\:\mathrm{cos}\:{x}\:\mathrm{log}\:\left(\frac{\mathrm{1}+{x}}{\mathrm{1}−{x}}\right)\:{dx}\:= \\ $$

Question Number 55773 Answers: 1 Comments: 0

∫_(−π) ^π sin x f(cos x) dx =

$$\:\underset{−\pi} {\overset{\pi} {\int}}\:\mathrm{sin}\:{x}\:{f}\left(\mathrm{cos}\:{x}\right)\:{dx}\:= \\ $$

Question Number 55772 Answers: 1 Comments: 1

∫_(−π/3) ^(π/3) ((x sin x)/(cos^2 x)) dx =

$$\:\underset{−\pi/\mathrm{3}} {\overset{\pi/\mathrm{3}} {\int}}\:\frac{{x}\:\mathrm{sin}\:{x}}{\mathrm{cos}^{\mathrm{2}} {x}}\:{dx}\:= \\ $$

Question Number 55771 Answers: 0 Comments: 0

Find all possible solutions (a, b, c) from Diophantine equation 2^a + 5^b = c^2

$$\mathrm{Find}\:\mathrm{all}\:\mathrm{possible}\:\mathrm{solutions}\:\left({a},\:{b},\:{c}\right)\:\mathrm{from} \\ $$$$\mathrm{Diophantine}\:\mathrm{equation}\:\:\mathrm{2}^{{a}} \:+\:\mathrm{5}^{{b}} \:=\:{c}^{\mathrm{2}} \\ $$

Question Number 55770 Answers: 0 Comments: 0

Let A and B are matrices in R^(2017×2017) that satisfy A^(−1) = (A + B)^(−1) − B^(−1) and det(A^(−1) ) = 2017 Find det(B)

$$\mathrm{Let}\:{A}\:\mathrm{and}\:{B}\:\mathrm{are}\:\mathrm{matrices}\:\mathrm{in}\:\mathbb{R}^{\mathrm{2017}×\mathrm{2017}} \:\mathrm{that}\:\mathrm{satisfy} \\ $$$${A}^{−\mathrm{1}} \:=\:\left({A}\:+\:{B}\right)^{−\mathrm{1}} \:−\:{B}^{−\mathrm{1}} \\ $$$$\mathrm{and} \\ $$$$\mathrm{det}\left({A}^{−\mathrm{1}} \right)\:=\:\mathrm{2017} \\ $$$$\mathrm{Find}\:\:\:\mathrm{det}\left({B}\right) \\ $$

Question Number 55762 Answers: 0 Comments: 0

Question Number 55760 Answers: 0 Comments: 3

let f(x) =∫_0 ^∞ ((cos(xt))/((xt^2 +i)^2 ))dx with x from R and x≠0 1) find a explicit form of f(x) 2) extract A =Re(f(x)) and B =Im(f(x)) and find its values . 3) calculate ∫_0 ^∞ ((cos(2t))/((2t^2 +i)^2 ))dt 4) let U_n =∫_0 ^∞ ((cos(nt))/((nt^2 +i)^2 ))dt .calculate lim_(n→+∞) u_n and study the convergence of Σu_n

$${let}\:{f}\left({x}\right)\:=\int_{\mathrm{0}} ^{\infty} \:\:\:\:\frac{{cos}\left({xt}\right)}{\left({xt}^{\mathrm{2}} +{i}\right)^{\mathrm{2}} }{dx}\:\:\:{with}\:{x}\:{from}\:{R}\:\:{and}\:{x}\neq\mathrm{0} \\ $$$$\left.\mathrm{1}\right)\:{find}\:{a}\:{explicit}\:{form}\:{of}\:{f}\left({x}\right) \\ $$$$\left.\mathrm{2}\right)\:{extract}\:\:{A}\:={Re}\left({f}\left({x}\right)\right)\:{and}\:\:{B}\:={Im}\left({f}\left({x}\right)\right)\:{and}\:{find}\:{its}\:{values}\:. \\ $$$$\left.\mathrm{3}\right)\:{calculate}\:\int_{\mathrm{0}} ^{\infty} \:\:\:\frac{{cos}\left(\mathrm{2}{t}\right)}{\left(\mathrm{2}{t}^{\mathrm{2}} \:+{i}\right)^{\mathrm{2}} }{dt} \\ $$$$\left.\mathrm{4}\right)\:{let}\:{U}_{{n}} =\int_{\mathrm{0}} ^{\infty} \:\:\frac{{cos}\left({nt}\right)}{\left({nt}^{\mathrm{2}} \:+{i}\right)^{\mathrm{2}} }{dt}\:\:\:.{calculate}\:{lim}_{{n}\rightarrow+\infty} \:{u}_{{n}} \\ $$$${and}\:{study}\:{the}\:{convergence}\:{of}\:\Sigma{u}_{{n}} \\ $$$$ \\ $$

Question Number 55759 Answers: 1 Comments: 0

calculate I =∫_0 ^(2π) ((cost)/(3 +sin(2t)))dt and J =∫_0 ^(2π) ((sint)/(3 +cos(2t)))dt .

$${calculate}\:{I}\:=\int_{\mathrm{0}} ^{\mathrm{2}\pi} \:\:\frac{{cost}}{\mathrm{3}\:+{sin}\left(\mathrm{2}{t}\right)}{dt}\:{and}\:{J}\:=\int_{\mathrm{0}} ^{\mathrm{2}\pi} \:\:\frac{{sint}}{\mathrm{3}\:+{cos}\left(\mathrm{2}{t}\right)}{dt}\:. \\ $$

Question Number 55756 Answers: 1 Comments: 0

Find the product of all the real values of a that satisfies the equation 4∣a−4∣=∣a+4∣

$$\mathrm{Find}\:\mathrm{the}\:\mathrm{product}\:\mathrm{of}\:\mathrm{all}\:\mathrm{the}\:\mathrm{real}\:\mathrm{values} \\ $$$$\mathrm{of}\:\:\:\mathrm{a}\:\:\mathrm{that}\:\mathrm{satisfies}\:\mathrm{the}\:\mathrm{equation} \\ $$$$\mathrm{4}\mid\mathrm{a}−\mathrm{4}\mid=\mid\mathrm{a}+\mathrm{4}\mid \\ $$

Question Number 55748 Answers: 1 Comments: 1

Question Number 55737 Answers: 1 Comments: 2

Question Number 55736 Answers: 1 Comments: 0

Question Number 55735 Answers: 1 Comments: 0

Question Number 55733 Answers: 0 Comments: 1

Question Number 55725 Answers: 1 Comments: 0

Question Number 55724 Answers: 0 Comments: 4

Question Number 55707 Answers: 0 Comments: 0

Question Number 55706 Answers: 0 Comments: 0

Question Number 55705 Answers: 0 Comments: 0

Question Number 55704 Answers: 1 Comments: 0

Prove that: 2 sin (1/2)θcos (3/2)θ+2sin (5/2)θ +2 sin (3/2)θ+2sin (3/2)θcos (7/2)θ =sin 4θ+sin 5θ

$$\mathrm{Prove}\:\mathrm{that}: \\ $$$$\mathrm{2}\:\mathrm{sin}\:\frac{\mathrm{1}}{\mathrm{2}}\theta\mathrm{cos}\:\frac{\mathrm{3}}{\mathrm{2}}\theta+\mathrm{2sin}\:\frac{\mathrm{5}}{\mathrm{2}}\theta\: \\ $$$$+\mathrm{2}\:\mathrm{sin}\:\frac{\mathrm{3}}{\mathrm{2}}\theta+\mathrm{2sin}\:\frac{\mathrm{3}}{\mathrm{2}}\theta\mathrm{cos}\:\frac{\mathrm{7}}{\mathrm{2}}\theta \\ $$$$=\mathrm{sin}\:\mathrm{4}\theta+\mathrm{sin}\:\mathrm{5}\theta \\ $$

Question Number 55702 Answers: 0 Comments: 5

s=∫_0 ^( x) (√(1+(3t^2 +p)^2 ))dt = ? take p=1 for a special case.

$${s}=\int_{\mathrm{0}} ^{\:{x}} \sqrt{\mathrm{1}+\left(\mathrm{3}{t}^{\mathrm{2}} +{p}\right)^{\mathrm{2}} }{dt}\:\:=\:? \\ $$$$\:\:\:\:{take}\:{p}=\mathrm{1}\:\:{for}\:{a}\:{special}\:{case}. \\ $$

Question Number 55701 Answers: 1 Comments: 1

Is true or not that 4181 is the only one Fibonacci′s number with no prime factor which is also a Fibonacci′s number?

$${Is}\:{true}\:{or}\:{not}\:{that}\:\mathrm{4181}\:{is}\:{the}\:{only}\:{one} \\ $$$${Fibonacci}'{s}\:{number}\:{with}\:{no}\:{prime}\:{factor} \\ $$$${which}\:{is}\:{also}\:{a}\:{Fibonacci}'{s}\:{number}? \\ $$

Question Number 55685 Answers: 1 Comments: 0

proof that Σ_(i=1) ^n (a_i /(a_i −x))=2015 has exactly n real roots.o<a_1 ....<a_n

$${proof}\:{that}\: \\ $$$$\underset{{i}=\mathrm{1}} {\overset{{n}} {\sum}}\frac{{a}_{{i}} }{{a}_{{i}} −{x}}=\mathrm{2015}\:{has}\:{exactly}\:{n}\:{real}\: \\ $$$${roots}.{o}<{a}_{\mathrm{1}} ....<{a}_{{n}} \\ $$

Question Number 55675 Answers: 0 Comments: 0

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