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Question Number 192688    Answers: 1   Comments: 0

Prove that : C_n ^k = (1/(2π)) ∫^( π) _( −π) (2cos(θ/2))^n cos[((n/2)−k)θ]dθ

$$\mathrm{Prove}\:\mathrm{that}\:: \\ $$$$\mathrm{C}_{\mathrm{n}} ^{\mathrm{k}} \:=\:\frac{\mathrm{1}}{\mathrm{2}\pi}\:\underset{\:−\pi} {\int}^{\:\:\:\pi} \left(\mathrm{2cos}\frac{\theta}{\mathrm{2}}\right)^{\mathrm{n}} \mathrm{cos}\left[\left(\frac{\mathrm{n}}{\mathrm{2}}−\mathrm{k}\right)\theta\right]\mathrm{d}\theta \\ $$

Question Number 192687    Answers: 1   Comments: 0

(1/π)∫_0 ^(2π) xcos(nx)dx Help!

$$\frac{\mathrm{1}}{\pi}\int_{\mathrm{0}} ^{\mathrm{2}\pi} \mathrm{xcos}\left(\mathrm{nx}\right)\mathrm{dx} \\ $$$$ \\ $$$$\mathrm{Help}! \\ $$

Question Number 192676    Answers: 1   Comments: 0

Find: x = ? 1. lg(5x^2 − 6)∙lg(5x − 6) = 0 2. (2x − 5)∙log_3 (1,5 − x) = 0 3. 4^x − 14∙2^x − 32 = 0

$$\mathrm{Find}:\:\:\:\mathrm{x}\:=\:? \\ $$$$\mathrm{1}.\:\:\mathrm{lg}\left(\mathrm{5x}^{\mathrm{2}} \:−\:\mathrm{6}\right)\centerdot\mathrm{lg}\left(\mathrm{5x}\:−\:\mathrm{6}\right)\:=\:\mathrm{0} \\ $$$$\mathrm{2}.\:\:\left(\mathrm{2x}\:−\:\mathrm{5}\right)\centerdot\mathrm{log}_{\mathrm{3}} \left(\mathrm{1},\mathrm{5}\:−\:\mathrm{x}\right)\:=\:\mathrm{0} \\ $$$$\mathrm{3}.\:\:\mathrm{4}^{\boldsymbol{\mathrm{x}}} \:−\:\mathrm{14}\centerdot\mathrm{2}^{\boldsymbol{\mathrm{x}}} \:−\:\mathrm{32}\:=\:\mathrm{0} \\ $$

Question Number 192675    Answers: 1   Comments: 0

Question Number 192668    Answers: 2   Comments: 0

∫_0 ^4 (1/x)e^x dx − ∫_0 ^4 (1/x)e^((1/2)x) dx= ?

$$\int_{\mathrm{0}} ^{\mathrm{4}} \frac{\mathrm{1}}{{x}}{e}^{{x}} {dx}\:−\:\int_{\mathrm{0}} ^{\mathrm{4}} \frac{\mathrm{1}}{{x}}{e}^{\frac{\mathrm{1}}{\mathrm{2}}{x}} {dx}=\:? \\ $$

Question Number 192664    Answers: 0   Comments: 0

Question Number 192663    Answers: 1   Comments: 0

Question Number 192661    Answers: 1   Comments: 2

Question Number 192652    Answers: 2   Comments: 0

lim_(x→0) ((1−(√(cos(x))))/(1+cos((√x))))

$${lim}_{{x}\rightarrow\mathrm{0}} \frac{\mathrm{1}−\sqrt{{cos}\left({x}\right)}}{\mathrm{1}+{cos}\left(\sqrt{{x}}\right)} \\ $$

Question Number 192651    Answers: 2   Comments: 0

lim_(x→0) ((2−(√(cos(x)))−cos(x))/x^2 )

$$ \\ $$$${lim}_{{x}\rightarrow\mathrm{0}} \frac{\mathrm{2}−\sqrt{{cos}\left({x}\right)}−{cos}\left({x}\right)}{{x}^{\mathrm{2}} } \\ $$

Question Number 192650    Answers: 2   Comments: 0

lim_(x→0) ((x−sen(x))/(tan^3 (x))) without lhopital rule

$$ \\ $$$$\boldsymbol{\mathrm{lim}}_{\boldsymbol{\mathrm{x}}\rightarrow\mathrm{0}} \frac{{x}−{sen}\left({x}\right)}{{tan}^{\mathrm{3}} \left({x}\right)} \\ $$$${without}\:{lhopital}\:{rule} \\ $$

Question Number 192648    Answers: 1   Comments: 0

find g(f(x)) f(x)=x^3 −x^2 +3x g(x)=x^2 −2x+1

$$\boldsymbol{\mathrm{find}}\:\boldsymbol{\mathrm{g}}\left(\boldsymbol{\mathrm{f}}\left(\boldsymbol{\mathrm{x}}\right)\right) \\ $$$$\boldsymbol{\mathrm{f}}\left(\boldsymbol{\mathrm{x}}\right)=\boldsymbol{\mathrm{x}}^{\mathrm{3}} −\boldsymbol{\mathrm{x}}^{\mathrm{2}} +\mathrm{3}\boldsymbol{\mathrm{x}} \\ $$$$\boldsymbol{\mathrm{g}}\left(\boldsymbol{\mathrm{x}}\right)=\boldsymbol{\mathrm{x}}^{\mathrm{2}} −\mathrm{2}\boldsymbol{\mathrm{x}}+\mathrm{1} \\ $$

Question Number 192646    Answers: 1   Comments: 0

lim_(x⇒3) (√((x^2 −4x+3)/(x−3))) find the limit

$$\boldsymbol{\mathrm{lim}}_{\boldsymbol{\mathrm{x}}\Rightarrow\mathrm{3}} \sqrt{\frac{\boldsymbol{\mathrm{x}}^{\mathrm{2}} −\mathrm{4}\boldsymbol{\mathrm{x}}+\mathrm{3}}{\boldsymbol{\mathrm{x}}−\mathrm{3}}} \\ $$$$\boldsymbol{\mathrm{find}}\:\boldsymbol{\mathrm{the}}\:\boldsymbol{\mathrm{limit}} \\ $$

Question Number 192639    Answers: 2   Comments: 0

lim_(x⇒∝ ) ((3x^3 +3x^2 +1)/(5x^3 +4x^2 +x)) find the limit

$$ \\ $$lim_(x⇒∝ ) ((3x^3 +3x^2 +1)/(5x^3 +4x^2 +x)) find the limit

Question Number 192636    Answers: 1   Comments: 0

lim_(x→1 ) (3x^2 −7x+3)^(10) find the limit

$${lim}_{{x}\rightarrow\mathrm{1}\:} \left(\mathrm{3}{x}^{\mathrm{2}} \:−\mathrm{7}{x}+\mathrm{3}\right)^{\mathrm{10}} \: \\ $$$${find}\:{the}\:{limit} \\ $$

Question Number 192634    Answers: 1   Comments: 0

∫sin(12x +8 )dx

$$\int\boldsymbol{{sin}}\left(\mathrm{12}{x}\:+\mathrm{8}\:\right){dx} \\ $$

Question Number 192630    Answers: 0   Comments: 0

z=xy−5x+2y. find (dz/dx) and (dz/dy) at(2,4)

$${z}={xy}−\mathrm{5}{x}+\mathrm{2}{y}.\:{find}\:\frac{{dz}}{{dx}}\:{and}\:\frac{{dz}}{{dy}}\:{at}\left(\mathrm{2},\mathrm{4}\right) \\ $$

Question Number 192629    Answers: 0   Comments: 0

Z=f(x_(1,) x_(2,) x_3 )=x_1 x_2 +x_1 ^5 −x_2 ^2 x_3 find f_1 ,f_(11) ,and f_(21)

$${Z}={f}\left({x}_{\mathrm{1},} {x}_{\mathrm{2},} {x}_{\mathrm{3}} \right)={x}_{\mathrm{1}} {x}_{\mathrm{2}} +{x}_{\mathrm{1}} ^{\mathrm{5}} −{x}_{\mathrm{2}} ^{\mathrm{2}} {x}_{\mathrm{3}} \:{find}\:{f}_{\mathrm{1}} ,{f}_{\mathrm{11}} ,{and}\:{f}_{\mathrm{21}} \\ $$

Question Number 192625    Answers: 3   Comments: 1

lim_(x→0) ((sin^4 (πcos(x)))/(1−cos(1−cos(1−cos(x)))))

$${lim}_{{x}\rightarrow\mathrm{0}} \frac{{sin}^{\mathrm{4}} \left(\pi{cos}\left({x}\right)\right)}{\mathrm{1}−{cos}\left(\mathrm{1}−{cos}\left(\mathrm{1}−{cos}\left({x}\right)\right)\right)} \\ $$

Question Number 192624    Answers: 1   Comments: 0

cos36−cos72=?

$${cos}\mathrm{36}−{cos}\mathrm{72}=? \\ $$

Question Number 192623    Answers: 1   Comments: 0

cot70+4cos70=?

$${cot}\mathrm{70}+\mathrm{4}{cos}\mathrm{70}=? \\ $$

Question Number 192622    Answers: 1   Comments: 0

(3/2)cos^(−1) (√(2/(2+π^2 )))+(1/4)sin^(−1) ((2(√(2π)))/(2+π^2 ))+tan^(−1) ((√2)/π)

$$\frac{\mathrm{3}}{\mathrm{2}}\mathrm{cos}^{−\mathrm{1}} \sqrt{\frac{\mathrm{2}}{\mathrm{2}+\pi^{\mathrm{2}} }}+\frac{\mathrm{1}}{\mathrm{4}}\mathrm{sin}^{−\mathrm{1}} \frac{\mathrm{2}\sqrt{\mathrm{2}\pi}}{\mathrm{2}+\pi^{\mathrm{2}} }+\mathrm{tan}^{−\mathrm{1}} \frac{\sqrt{\mathrm{2}}}{\pi} \\ $$

Question Number 192617    Answers: 2   Comments: 0

Question Number 192616    Answers: 0   Comments: 0

Question Number 192612    Answers: 1   Comments: 3

(1/x) + (1/y) + (1/z) = 1 find the minimum value of x^2 + y^2 + z^2

$$\frac{\mathrm{1}}{{x}}\:+\:\frac{\mathrm{1}}{{y}}\:+\:\frac{\mathrm{1}}{{z}}\:=\:\mathrm{1}\: \\ $$$${find}\:{the}\:{minimum}\:{value}\:{of}\:{x}^{\mathrm{2}} \:+\:{y}^{\mathrm{2}} \:+\:{z}^{\mathrm{2}} \\ $$

Question Number 192608    Answers: 2   Comments: 0

let k be natural number. defined s_k as the sum of the infinite series s_k =((k^2 −1)/k^0 ) + ((k^2 −1)/k^1 ) + ((k^2 −1)/k^2 ) +... find the value of Σ_(k=1) ^∞ [(s_k /2^(k−1) )] .

$${let}\:{k}\:{be}\:{natural}\:{number}.\:{defined}\:{s}_{{k}} \:{as}\:{the} \\ $$$${sum}\:{of}\:{the}\:{infinite}\:{series}\:{s}_{{k}} =\frac{{k}^{\mathrm{2}} −\mathrm{1}}{{k}^{\mathrm{0}} }\:+\:\frac{{k}^{\mathrm{2}} −\mathrm{1}}{{k}^{\mathrm{1}} }\:+\:\frac{{k}^{\mathrm{2}} −\mathrm{1}}{{k}^{\mathrm{2}} }\:+... \\ $$$${find}\:{the}\:{value}\:{of}\:\underset{{k}=\mathrm{1}} {\overset{\infty} {\sum}}\:\left[\frac{{s}_{{k}} }{\mathrm{2}^{{k}−\mathrm{1}} }\right]\:\:. \\ $$

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