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Question Number 84982 Answers: 5 Comments: 0

Σ_(n=2) ^∞ ((H_n H_(n+1) )/(n^3 −n))

$$\underset{{n}=\mathrm{2}} {\overset{\infty} {\sum}}\frac{{H}_{{n}} \:{H}_{{n}+\mathrm{1}} }{{n}^{\mathrm{3}} −{n}} \\ $$

Question Number 84970 Answers: 1 Comments: 0

Question Number 84969 Answers: 0 Comments: 0

let c is a constant vector and r^→ =xi^ +yj^ +zk^ then proved that grad ∣c×r^→ ∣^n =n∣c×r^→ ∣^(n−2) c×(r^→ ×c).

$$\mathrm{let}\:\mathrm{c}\:\mathrm{is}\:\mathrm{a}\:\mathrm{constant}\:\mathrm{vector}\:\mathrm{and}\:\overset{\rightarrow} {\mathrm{r}}=\mathrm{x}\hat {\mathrm{i}}+\mathrm{y}\hat {\mathrm{j}}+\mathrm{z}\hat {\mathrm{k}}\:\mathrm{then}\:\mathrm{proved}\:\mathrm{that}\:\mathrm{grad}\:\mid\mathrm{c}×\overset{\rightarrow} {\mathrm{r}}\mid^{\mathrm{n}} =\mathrm{n}\mid\mathrm{c}×\overset{\rightarrow} {\mathrm{r}}\mid^{\mathrm{n}−\mathrm{2}} \mathrm{c}×\left(\overset{\rightarrow} {\mathrm{r}}×\mathrm{c}\right). \\ $$

Question Number 84960 Answers: 2 Comments: 0

Question Number 84958 Answers: 0 Comments: 0

Question Number 84957 Answers: 0 Comments: 1

∫ (2−x^2 )^3 dx =

$$\int\:\left(\mathrm{2}−\mathrm{x}^{\mathrm{2}} \right)^{\mathrm{3}} \:\mathrm{dx}\:=\: \\ $$

Question Number 84956 Answers: 1 Comments: 3

show that ∫_0 ^(+∞) (1/(x^4 +2x^2 cos(((2π)/5))+1)) dx=(π/(2φ))

$${show}\:{that}\: \\ $$$$\int_{\mathrm{0}} ^{+\infty} \frac{\mathrm{1}}{{x}^{\mathrm{4}} +\mathrm{2}{x}^{\mathrm{2}} {cos}\left(\frac{\mathrm{2}\pi}{\mathrm{5}}\right)+\mathrm{1}}\:{dx}=\frac{\pi}{\mathrm{2}\phi} \\ $$

Question Number 84954 Answers: 1 Comments: 0

lim_(x→0) (((√x) − (√(sin x)))/x^(5/2) ) = ?

$$\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\frac{\sqrt{\mathrm{x}}\:−\:\sqrt{\mathrm{sin}\:\mathrm{x}}}{\mathrm{x}^{\frac{\mathrm{5}}{\mathrm{2}}} }\:=\:? \\ $$

Question Number 84976 Answers: 0 Comments: 4

Question Number 84932 Answers: 0 Comments: 1

lim_(x→0) ((sin 38x−38sin x)/(19x^3 )) =

$$\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\frac{\mathrm{sin}\:\mathrm{38x}−\mathrm{38sin}\:\mathrm{x}}{\mathrm{19x}^{\mathrm{3}} }\:=\: \\ $$

Question Number 84942 Answers: 1 Comments: 0

∫_0 ^x sinh(x−t) cosh(t) dt

$$\int_{\mathrm{0}} ^{{x}} {sinh}\left({x}−{t}\right)\:{cosh}\left({t}\right)\:{dt} \\ $$

Question Number 84941 Answers: 2 Comments: 1

Question Number 84915 Answers: 1 Comments: 0

if x>0,y>0,z>0 show that ((x+y)/z)+((z+y)/( x))+((z+x)/y)≥6

$${if}\: \\ $$$${x}>\mathrm{0},{y}>\mathrm{0},{z}>\mathrm{0} \\ $$$${show}\:{that} \\ $$$$\frac{{x}+{y}}{{z}}+\frac{{z}+{y}}{\:{x}}+\frac{{z}+{x}}{{y}}\geqslant\mathrm{6}\:\: \\ $$

Question Number 84913 Answers: 2 Comments: 2

sin(π/(14)) sin((3π)/(14)) sin((5π)/(15))=?

$$ \\ $$$${sin}\frac{\pi}{\mathrm{14}}\:{sin}\frac{\mathrm{3}\pi}{\mathrm{14}}\:{sin}\frac{\mathrm{5}\pi}{\mathrm{15}}=? \\ $$

Question Number 84909 Answers: 2 Comments: 5

Find all solutions of (x, y) such that x^3 − 3xy^2 = 2010 y^3 − 3x^2 y = 2009 x, y ∈ R

$${Find}\:\:\:{all}\:\:{solutions}\:\:{of}\:\:\left({x},\:{y}\right)\:\:{such}\:\:{that} \\ $$$$\:\:\:\:\:\:\:\:{x}^{\mathrm{3}} \:−\:\mathrm{3}{xy}^{\mathrm{2}} \:\:=\:\:\mathrm{2010} \\ $$$$\:\:\:\:\:\:\:\:{y}^{\mathrm{3}} \:−\:\mathrm{3}{x}^{\mathrm{2}} {y}\:\:=\:\:\mathrm{2009} \\ $$$${x},\:{y}\:\:\in\:\:\mathbb{R} \\ $$

Question Number 84904 Answers: 0 Comments: 1

3^(2x^2 ) + 3^(x^2 +2x+5) ≥ 10. 3^(4x+6)

$$\mathrm{3}^{\mathrm{2x}^{\mathrm{2}} } \:+\:\mathrm{3}^{\mathrm{x}^{\mathrm{2}} +\mathrm{2x}+\mathrm{5}} \:\geqslant\:\mathrm{10}.\:\mathrm{3}^{\mathrm{4x}+\mathrm{6}} \\ $$$$ \\ $$

Question Number 84902 Answers: 0 Comments: 2

lim_(x→∞) (5^x +5^(2x) )^(1/x) ?

$$\underset{{x}\rightarrow\infty} {\mathrm{lim}}\:\left(\mathrm{5}^{\mathrm{x}} +\mathrm{5}^{\mathrm{2x}} \right)\:^{\frac{\mathrm{1}}{\mathrm{x}}} \:? \\ $$

Question Number 84899 Answers: 0 Comments: 1

Question Number 84894 Answers: 0 Comments: 1

Question Number 84892 Answers: 1 Comments: 4

Question Number 84891 Answers: 0 Comments: 0

Question Number 84890 Answers: 0 Comments: 3

If we have : y = e^x What is : (d/dy)e^x = ... If we derivate with y... Please...

$$\mathrm{If}\:\mathrm{we}\:\mathrm{have}\::\:\:\:\:\:{y}\:=\:{e}^{{x}} \\ $$$$ \\ $$$${W}\mathrm{hat}\:\mathrm{is}\::\:\:\:\frac{\mathrm{d}}{\mathrm{d}{y}}{e}^{{x}} \:=\:... \\ $$$$ \\ $$$$\mathrm{If}\:\mathrm{we}\:\mathrm{derivate}\:\mathrm{with}\:{y}... \\ $$$$ \\ $$$$\mathrm{Please}... \\ $$

Question Number 84884 Answers: 2 Comments: 1

Question Number 84879 Answers: 2 Comments: 1

e^(∫((2dx)/(xlnx)))

$$\mathrm{e}^{\int\frac{\mathrm{2dx}}{\mathrm{xlnx}}} \\ $$

Question Number 84873 Answers: 2 Comments: 1

lim_(x→∞) ((x^2 sin (((x!)/x)))/(x^2 +1))

$$\underset{{x}\rightarrow\infty} {\mathrm{lim}}\:\frac{\mathrm{x}^{\mathrm{2}} \mathrm{sin}\:\left(\frac{\mathrm{x}!}{\mathrm{x}}\right)}{\mathrm{x}^{\mathrm{2}} +\mathrm{1}} \\ $$

Question Number 84871 Answers: 1 Comments: 1

If you know (((b^2 +c^2 −a^2 )/(2bc)))^2 +(((c^2 +a^2 −b^2 )/(2ca)))^2 +(((a^2 +b^2 −c^2 )/(2ab)))^2 =3, then what′s the value of ((b^2 +c^2 −a^2 )/(2bc))+((c^2 +a^2 −b^2 )/(2ac))+((a^2 +b^2 −c^2 )/(2ab))?

$$\mathrm{If}\:\mathrm{you}\:\mathrm{know} \\ $$$$\left(\frac{{b}^{\mathrm{2}} +{c}^{\mathrm{2}} −{a}^{\mathrm{2}} }{\mathrm{2}{bc}}\right)^{\mathrm{2}} +\left(\frac{{c}^{\mathrm{2}} +{a}^{\mathrm{2}} −{b}^{\mathrm{2}} }{\mathrm{2}{ca}}\right)^{\mathrm{2}} +\left(\frac{{a}^{\mathrm{2}} +{b}^{\mathrm{2}} −{c}^{\mathrm{2}} }{\mathrm{2}{ab}}\right)^{\mathrm{2}} =\mathrm{3}, \\ $$$$\mathrm{then}\:\mathrm{what}'\mathrm{s}\:\mathrm{the}\:\mathrm{value}\:\mathrm{of} \\ $$$$\frac{{b}^{\mathrm{2}} +{c}^{\mathrm{2}} −{a}^{\mathrm{2}} }{\mathrm{2}{bc}}+\frac{{c}^{\mathrm{2}} +{a}^{\mathrm{2}} −{b}^{\mathrm{2}} }{\mathrm{2}{ac}}+\frac{{a}^{\mathrm{2}} +{b}^{\mathrm{2}} −{c}^{\mathrm{2}} }{\mathrm{2}{ab}}? \\ $$

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