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Question Number 157209    Answers: 2   Comments: 2

Question Number 157207    Answers: 0   Comments: 1

sin lnxdx

$$\mathrm{sin}\:\mathrm{ln}{xdx} \\ $$

Question Number 157141    Answers: 1   Comments: 1

{ (((d^2 y/dx^2 )−2 (dy/dx)+y =3e^(4x) )),((y(0)=−(2/3) ; (dy/dx)∣_(x=0) = ((13)/3))) :}

$$\:\begin{cases}{\frac{{d}^{\mathrm{2}} {y}}{{dx}^{\mathrm{2}} }−\mathrm{2}\:\frac{{dy}}{{dx}}+{y}\:=\mathrm{3}{e}^{\mathrm{4}{x}} }\\{{y}\left(\mathrm{0}\right)=−\frac{\mathrm{2}}{\mathrm{3}}\:;\:\frac{{dy}}{{dx}}\mid_{{x}=\mathrm{0}} =\:\frac{\mathrm{13}}{\mathrm{3}}}\end{cases}\: \\ $$

Question Number 157139    Answers: 1   Comments: 1

The figure given below is the section of the reinforced concrete short column. Calculate the stress in the concrete and the stress in the steel,if the axial load of 780KN is applied to the column. Young′s modulus of steel=210KN/mm^2 Young′s modulus of concrete=14KN/mm^2 diameter of the steel bars=20mm

$${The}\:{figure}\:{given}\:{below}\:{is}\:{the}\:{section} \\ $$$${of}\:{the}\:{reinforced}\:{concrete}\:{short}\:{column}. \\ $$$${Calculate}\:{the}\:{stress}\:{in}\:{the}\:{concrete} \\ $$$${and}\:{the}\:{stress}\:{in}\:{the}\:{steel},{if}\:{the} \\ $$$${axial}\:{load}\:{of}\:\mathrm{780}{KN}\:{is}\:{applied}\:{to} \\ $$$${the}\:{column}. \\ $$$$ \\ $$$${Young}'{s}\:{modulus}\:{of}\:{steel}=\mathrm{210}{KN}/{mm}^{\mathrm{2}} \\ $$$${Young}'{s}\:{modulus}\:{of}\:{concrete}=\mathrm{14}{KN}/{mm}^{\mathrm{2}} \\ $$$${diameter}\:{of}\:{the}\:{steel}\:{bars}=\mathrm{20}{mm} \\ $$

Question Number 157138    Answers: 0   Comments: 4

∫_0 ^1 x^x (1−x)^(1−x) sin(πx)dx=?

$$\int_{\mathrm{0}} ^{\mathrm{1}} \boldsymbol{\mathrm{x}}^{\boldsymbol{\mathrm{x}}} \left(\mathrm{1}−\boldsymbol{\mathrm{x}}\right)^{\mathrm{1}−\boldsymbol{\mathrm{x}}} \boldsymbol{\mathrm{sin}}\left(\pi\boldsymbol{\mathrm{x}}\right)\boldsymbol{\mathrm{dx}}=? \\ $$

Question Number 157126    Answers: 0   Comments: 9

how many ways can you exchange 1 dollar using (1,5,10,25,50) cents example 1 dollar(100 cents)=2×50cents =1×50+2×25=4×25 =1×50+1×25+1×10+1×5+10×1 and so on how many all the ways ?

$${how}\:{many}\:{ways}\:{can}\:{you} \\ $$$${exchange}\:\mathrm{1}\:{dollar}\:{using} \\ $$$$\left(\mathrm{1},\mathrm{5},\mathrm{10},\mathrm{25},\mathrm{50}\right)\:{cents} \\ $$$${example} \\ $$$$\mathrm{1}\:{dollar}\left(\mathrm{100}\:{cents}\right)=\mathrm{2}×\mathrm{50}{cents} \\ $$$$=\mathrm{1}×\mathrm{50}+\mathrm{2}×\mathrm{25}=\mathrm{4}×\mathrm{25} \\ $$$$=\mathrm{1}×\mathrm{50}+\mathrm{1}×\mathrm{25}+\mathrm{1}×\mathrm{10}+\mathrm{1}×\mathrm{5}+\mathrm{10}×\mathrm{1} \\ $$$${and}\:{so}\:{on} \\ $$$${how}\:{many}\:{all}\:{the}\:{ways}\:? \\ $$

Question Number 157123    Answers: 0   Comments: 1

lim_(x→0) ((log _(sin x) (cos x))/(log _(sin ((x/2))) (cos (x/2))))=?

$$\:\:\:\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\frac{\mathrm{log}\:_{\mathrm{sin}\:{x}} \left(\mathrm{cos}\:{x}\right)}{\mathrm{log}\:_{\mathrm{sin}\:\left(\frac{{x}}{\mathrm{2}}\right)} \left(\mathrm{cos}\:\frac{{x}}{\mathrm{2}}\right)}=? \\ $$

Question Number 157120    Answers: 2   Comments: 0

Σ_(n=0) ^∞ ((1/((3n+1)(3n+2))))=?

$$\:\:\:\underset{{n}=\mathrm{0}} {\overset{\infty} {\sum}}\left(\frac{\mathrm{1}}{\left(\mathrm{3}{n}+\mathrm{1}\right)\left(\mathrm{3}{n}+\mathrm{2}\right)}\right)=? \\ $$

Question Number 157116    Answers: 1   Comments: 2

max ∧ min of f(x) =(√x) +4(√((1−x)/2))

$$\:{max}\:\wedge\:{min}\:{of}\:{f}\left({x}\right)\:=\sqrt{{x}}\:+\mathrm{4}\sqrt{\frac{\mathrm{1}−{x}}{\mathrm{2}}} \\ $$

Question Number 157114    Answers: 0   Comments: 2

Question Number 157198    Answers: 1   Comments: 0

Montrer que l′ensemble Q n′est ni ferme ni ouvert

$$\mathrm{M}{ontrer}\:{que}\:{l}'{ensemble}\:{Q}\:{n}'{est}\:{ni}\:{ferme}\:{ni}\:{ouvert} \\ $$

Question Number 157197    Answers: 1   Comments: 0

Question Number 157196    Answers: 1   Comments: 0

Question Number 157104    Answers: 2   Comments: 0

prove lim_(△x→0) ((e^(△x) −1)/(△x))=1

$${prove}\:\underset{\bigtriangleup{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\frac{{e}^{\bigtriangleup{x}} −\mathrm{1}}{\bigtriangleup{x}}=\mathrm{1} \\ $$

Question Number 157099    Answers: 0   Comments: 2

Question Number 157098    Answers: 1   Comments: 0

n ∈ N^∗ ; n is not a square of any integer. Show that (√n) ∉ Q .

$${n}\:\in\:\mathbb{N}^{\ast} \:;\:{n}\:{is}\:{not}\:{a}\:{square}\:{of}\:{any} \\ $$$${integer}.\:{Show}\:{that}\:\sqrt{{n}}\:\notin\:{Q}\:. \\ $$

Question Number 157096    Answers: 1   Comments: 0

∫_0 ^(π/2) xsin(x)ln(sin(x))dx=?

$$\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \boldsymbol{\mathrm{xsin}}\left(\boldsymbol{\mathrm{x}}\right)\boldsymbol{\mathrm{ln}}\left(\boldsymbol{\mathrm{sin}}\left(\boldsymbol{\mathrm{x}}\right)\right)\boldsymbol{\mathrm{dx}}=? \\ $$

Question Number 157087    Answers: 0   Comments: 0

Let ABCD be a tetrahedron, G, G' and G" the centers of gravity of faces ABC, ACD and ADB, I and J the midpoints of edges [BC] and [CD]. 1) Show that lines (IJ) and (GG' ) are parallel. 2) Prove that the points G, G' and G" are not aligned. 3) Show that the planes (GG 'G ") and (BCD) are parallel.

$$ \\ $$Let ABCD be a tetrahedron, G, G' and G" the centers of gravity of faces ABC, ACD and ADB, I and J the midpoints of edges [BC] and [CD]. 1) Show that lines (IJ) and (GG' ) are parallel. 2) Prove that the points G, G' and G" are not aligned. 3) Show that the planes (GG 'G ") and (BCD) are parallel.

Question Number 157083    Answers: 0   Comments: 0

Question Number 157080    Answers: 3   Comments: 2

Question Number 157074    Answers: 2   Comments: 0

(√(4+(√(4^2 +(√(4^3 +(√(4^4 +…))))))))=?

$$\sqrt{\mathrm{4}+\sqrt{\mathrm{4}^{\mathrm{2}} +\sqrt{\mathrm{4}^{\mathrm{3}} +\sqrt{\mathrm{4}^{\mathrm{4}} +\ldots}}}}=? \\ $$

Question Number 157071    Answers: 0   Comments: 0

if 0<x≤(π/2) then: (1/(sinx)) + (2/(πx)) ≤ (1 - (2/π))^2 + (1/x) + (2/π)

$$\mathrm{if}\:\:\:\mathrm{0}<\mathrm{x}\leqslant\frac{\pi}{\mathrm{2}}\:\:\:\mathrm{then}: \\ $$$$\frac{\mathrm{1}}{\mathrm{sin}\boldsymbol{\mathrm{x}}}\:+\:\frac{\mathrm{2}}{\pi\mathrm{x}}\:\leqslant\:\left(\mathrm{1}\:-\:\frac{\mathrm{2}}{\pi}\right)^{\mathrm{2}} +\:\frac{\mathrm{1}}{\mathrm{x}}\:+\:\frac{\mathrm{2}}{\pi} \\ $$

Question Number 157065    Answers: 0   Comments: 1

prove that cos 36°=((1+(√5))/4)

$$\mathrm{prove}\:\mathrm{that}\:\mathrm{cos}\:\mathrm{36}°=\frac{\mathrm{1}+\sqrt{\mathrm{5}}}{\mathrm{4}} \\ $$

Question Number 157061    Answers: 2   Comments: 0

Show that ∀ n ∈ N, ⌊((√n)+(√(n+1)))^2 ⌋=4n+1

$$ \\ $$$${Show}\:{that}\:\forall\:{n}\:\in\:\mathbb{N},\: \\ $$$$\lfloor\left(\sqrt{{n}}+\sqrt{{n}+\mathrm{1}}\right)^{\mathrm{2}} \rfloor=\mathrm{4}{n}+\mathrm{1} \\ $$

Question Number 157055    Answers: 1   Comments: 1

Question Number 157053    Answers: 0   Comments: 2

If you want to easily write any quadratic function in vertex form, just use these formulas: f(x)=ax^2 +bx+c if a>0, then: f(x)=(x+(b/2))^2 −((b/2))^2 +c if a<0, then: f(x)=−(x−((b/2)))^2 +((b/2))^2 +c Let′s take some examples: f(x)=x^2 −6x+7 f(x)=(x−(6/2))^2 −((6/2))^2 +7 f(x)=(x−3)^2 −9+7 f(x)=(x−3)^2 −2 Now let′s take the same example, but when a<0: f(x)=−x^2 −6x+7 f(x)=−(x−(−(6/2)))^2 +((6/2))^2 +7 f(x)=−(x+3)^2 +9+7 f(x)=−(x+3)^2 +16 Another example: f(x)=x^2 +4x−5 f(x)=(x+(4/2))^2 −((4/2))^2 −5 f(x)=(x+2)^2 −4−5 f(x)=(x+2)^2 −9 Now let′s also see what happens when a<0: f(x)=−x^2 +4x−5 f(x)=−(x−((4/2)))^2 +((4/2))^2 −5 f(x)=−(x−2)^2 +4−5 f(x)=−(x−2)^2 −1

$$\mathrm{If}\:\mathrm{you}\:\mathrm{want}\:\mathrm{to}\:\mathrm{easily}\:\mathrm{write}\:\mathrm{any}\:\mathrm{quadratic}\:\mathrm{function}\:\mathrm{in}\:\mathrm{vertex}\:\mathrm{form},\:\mathrm{just}\:\mathrm{use}\:\mathrm{these}\:\mathrm{formulas}: \\ $$$$\: \\ $$$${f}\left({x}\right)={ax}^{\mathrm{2}} +{bx}+{c} \\ $$$$\: \\ $$$$\mathrm{if}\:{a}>\mathrm{0},\:\mathrm{then}: \\ $$$${f}\left({x}\right)=\left({x}+\frac{{b}}{\mathrm{2}}\right)^{\mathrm{2}} −\left(\frac{{b}}{\mathrm{2}}\right)^{\mathrm{2}} +{c} \\ $$$$\: \\ $$$$\mathrm{if}\:{a}<\mathrm{0},\:\mathrm{then}: \\ $$$${f}\left({x}\right)=−\left({x}−\left(\frac{{b}}{\mathrm{2}}\right)\right)^{\mathrm{2}} +\left(\frac{{b}}{\mathrm{2}}\right)^{\mathrm{2}} +{c} \\ $$$$\: \\ $$$$\mathrm{Let}'\mathrm{s}\:\mathrm{take}\:\mathrm{some}\:\mathrm{examples}: \\ $$$${f}\left({x}\right)={x}^{\mathrm{2}} −\mathrm{6}{x}+\mathrm{7} \\ $$$${f}\left({x}\right)=\left({x}−\frac{\mathrm{6}}{\mathrm{2}}\right)^{\mathrm{2}} −\left(\frac{\mathrm{6}}{\mathrm{2}}\right)^{\mathrm{2}} +\mathrm{7} \\ $$$${f}\left({x}\right)=\left({x}−\mathrm{3}\right)^{\mathrm{2}} −\mathrm{9}+\mathrm{7} \\ $$$${f}\left({x}\right)=\left({x}−\mathrm{3}\right)^{\mathrm{2}} −\mathrm{2} \\ $$$$\: \\ $$$$\mathrm{Now}\:\mathrm{let}'\mathrm{s}\:\mathrm{take}\:\mathrm{the}\:\mathrm{same}\:\mathrm{example},\:\mathrm{but}\:\mathrm{when}\:{a}<\mathrm{0}: \\ $$$$\: \\ $$$${f}\left({x}\right)=−{x}^{\mathrm{2}} −\mathrm{6}{x}+\mathrm{7} \\ $$$${f}\left({x}\right)=−\left({x}−\left(−\frac{\mathrm{6}}{\mathrm{2}}\right)\right)^{\mathrm{2}} +\left(\frac{\mathrm{6}}{\mathrm{2}}\right)^{\mathrm{2}} +\mathrm{7} \\ $$$${f}\left({x}\right)=−\left({x}+\mathrm{3}\right)^{\mathrm{2}} +\mathrm{9}+\mathrm{7} \\ $$$${f}\left({x}\right)=−\left({x}+\mathrm{3}\right)^{\mathrm{2}} +\mathrm{16} \\ $$$$\: \\ $$$$\mathrm{Another}\:\mathrm{example}: \\ $$$$\: \\ $$$${f}\left({x}\right)={x}^{\mathrm{2}} +\mathrm{4}{x}−\mathrm{5} \\ $$$${f}\left({x}\right)=\left({x}+\frac{\mathrm{4}}{\mathrm{2}}\right)^{\mathrm{2}} −\left(\frac{\mathrm{4}}{\mathrm{2}}\right)^{\mathrm{2}} −\mathrm{5} \\ $$$${f}\left({x}\right)=\left({x}+\mathrm{2}\right)^{\mathrm{2}} −\mathrm{4}−\mathrm{5} \\ $$$${f}\left({x}\right)=\left({x}+\mathrm{2}\right)^{\mathrm{2}} −\mathrm{9} \\ $$$$\: \\ $$$$\mathrm{Now}\:\mathrm{let}'\mathrm{s}\:\mathrm{also}\:\mathrm{see}\:\mathrm{what}\:\mathrm{happens}\:\mathrm{when}\:{a}<\mathrm{0}: \\ $$$$\: \\ $$$${f}\left({x}\right)=−{x}^{\mathrm{2}} +\mathrm{4}{x}−\mathrm{5} \\ $$$${f}\left({x}\right)=−\left({x}−\left(\frac{\mathrm{4}}{\mathrm{2}}\right)\right)^{\mathrm{2}} +\left(\frac{\mathrm{4}}{\mathrm{2}}\right)^{\mathrm{2}} −\mathrm{5} \\ $$$${f}\left({x}\right)=−\left({x}−\mathrm{2}\right)^{\mathrm{2}} +\mathrm{4}−\mathrm{5} \\ $$$${f}\left({x}\right)=−\left({x}−\mathrm{2}\right)^{\mathrm{2}} −\mathrm{1} \\ $$$$\: \\ $$

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