Question and Answers Forum

Prove that for every positive real numbers x, y, z and xyz = 1, hold (x + y + z)^2 ((1/x^2 ) + (1/y^2 ) + (1/z^2 )) ≥ 9 + 2(x^3 + y^3 + z^3 ) + 4((1/x^3 ) + (1/y^3 ) + (1/z^3 ))

$${Prove}\:\:{that}\:\:{for}\:\:{every}\:\:{positive}\:\:{real}\:\:{numbers}\:\:{x},\:{y},\:{z}\:\:{and}\:\:\:{xyz}\:\:=\:\:\mathrm{1},\:\:{hold} \\ $$$$\left({x}\:+\:{y}\:+\:{z}\right)^{\mathrm{2}} \left(\frac{\mathrm{1}}{{x}^{\mathrm{2}} }\:+\:\frac{\mathrm{1}}{{y}^{\mathrm{2}} }\:+\:\frac{\mathrm{1}}{{z}^{\mathrm{2}} }\right)\:\:\geqslant\:\:\mathrm{9}\:+\:\mathrm{2}\left({x}^{\mathrm{3}} \:+\:{y}^{\mathrm{3}} \:+\:{z}^{\mathrm{3}} \right)\:+\:\mathrm{4}\left(\frac{\mathrm{1}}{{x}^{\mathrm{3}} }\:+\:\frac{\mathrm{1}}{{y}^{\mathrm{3}} }\:+\:\frac{\mathrm{1}}{{z}^{\mathrm{3}} }\right)\: \\ $$

Question Number 34029 Answers: 1 Comments: 1

Number of integral values of x for which ((((π/2^(tan^(−1) x) )−4)(x−4)(x−10))/(x! − (x−1)!)) < 0

$$\boldsymbol{{N}}{umber}\:{of}\:{integral}\:{values}\:{of}\:{x}\:{for} \\ $$$${which}\: \\ $$$$\frac{\left(\frac{\pi}{\mathrm{2}^{\mathrm{tan}^{−\mathrm{1}} {x}} }−\mathrm{4}\right)\left({x}−\mathrm{4}\right)\left({x}−\mathrm{10}\right)}{{x}!\:−\:\left({x}−\mathrm{1}\right)!}\:<\:\mathrm{0} \\ $$

Question Number 34021 Answers: 1 Comments: 2

find the value of ∫_0 ^(+∞) ((cos(αx))/((x^2 +1)( x^2 +2)(x^2 +3)))dx 2) calculate ∫_0 ^∞ (dx/((x^2 +1)(x^2 +2)(x^2 +3)))

$${find}\:{the}\:{value}\:{of}\:\:\int_{\mathrm{0}} ^{+\infty} \:\:\:\frac{{cos}\left(\alpha{x}\right)}{\left({x}^{\mathrm{2}} +\mathrm{1}\right)\left(\:{x}^{\mathrm{2}} +\mathrm{2}\right)\left({x}^{\mathrm{2}} +\mathrm{3}\right)}{dx} \\ $$$$\left.\mathrm{2}\right)\:{calculate}\:\:\int_{\mathrm{0}} ^{\infty} \:\:\:\:\:\:\:\:\:\:\frac{{dx}}{\left({x}^{\mathrm{2}} +\mathrm{1}\right)\left({x}^{\mathrm{2}} \:+\mathrm{2}\right)\left({x}^{\mathrm{2}} +\mathrm{3}\right)} \\ $$

Question Number 34020 Answers: 0 Comments: 1

let p(x)=cos(2n arccos(x)) with x∈[−1,1] find the roots of p(x) and factorize p(x)

$${let}\:{p}\left({x}\right)={cos}\left(\mathrm{2}{n}\:{arccos}\left({x}\right)\right)\:\:{with}\:{x}\in\left[−\mathrm{1},\mathrm{1}\right] \\ $$$${find}\:{the}\:{roots}\:{of}\:{p}\left({x}\right)\:{and}\:{factorize}\:\:{p}\left({x}\right) \\ $$

Question Number 34019 Answers: 0 Comments: 4

n integr decompose imsidr R[x] the fraction F(x) = (1/((x^2 −1)^n ))

$${n}\:{integr}\:{decompose}\:{imsidr}\:{R}\left[{x}\right]\:{the}\:{fraction} \\ $$$${F}\left({x}\right)\:=\:\:\:\frac{\mathrm{1}}{\left({x}^{\mathrm{2}} \:−\mathrm{1}\right)^{{n}} } \\ $$

Question Number 34013 Answers: 0 Comments: 0

find Σ_(n=0) ^∞ ((cos((2n+1)(π/4)))/((2n+1)^2 )) .

$${find}\:\:\sum_{{n}=\mathrm{0}} ^{\infty} \:\:\:\:\frac{{cos}\left(\left(\mathrm{2}{n}+\mathrm{1}\right)\frac{\pi}{\mathrm{4}}\right)}{\left(\mathrm{2}{n}+\mathrm{1}\right)^{\mathrm{2}} }\:. \\ $$

Question Number 34011 Answers: 0 Comments: 0

calculate Σ_(n=1) ^∞ ((cos(nx))/n^2 ) and Σ_(n=1) ^∞ ((sin(nx))/n^2 )

$${calculate}\:\:\sum_{{n}=\mathrm{1}} ^{\infty} \:\frac{{cos}\left({nx}\right)}{{n}^{\mathrm{2}} }\:\:{and}\:\sum_{{n}=\mathrm{1}} ^{\infty} \:\:\frac{{sin}\left({nx}\right)}{{n}^{\mathrm{2}} } \\ $$

Question Number 34007 Answers: 1 Comments: 0

lim_(x→2) (((√(x−2)) +(√x) −(√2))/(√(x^2 −4))) is ?

$$\underset{{x}\rightarrow\mathrm{2}} {\mathrm{lim}}\frac{\sqrt{{x}−\mathrm{2}}\:+\sqrt{{x}}\:−\sqrt{\mathrm{2}}}{\sqrt{{x}^{\mathrm{2}} −\mathrm{4}}}\:\:{is}\:? \\ $$

Question Number 34005 Answers: 1 Comments: 4

((4k+1)/(k+3)),(4k+1,k+3)=(11,k+3)=1or11; I can′t understand.Who can help me?

$$\frac{\mathrm{4}{k}+\mathrm{1}}{{k}+\mathrm{3}},\left(\mathrm{4}{k}+\mathrm{1},{k}+\mathrm{3}\right)=\left(\mathrm{11},{k}+\mathrm{3}\right)=\mathrm{1}{or}\mathrm{11}; \\ $$$${I}\:{can}'{t}\:{understand}.{Who}\:{can}\:{help}\:{me}? \\ $$

Question Number 33997 Answers: 1 Comments: 0

If the range of the function f(x) = ((x−1)/(p−x^2 +1)) does not contain any values belonging to the interval [−1,((−1)/3)] then true set of values of p is ?

$$\boldsymbol{{I}}\mathrm{f}\:\mathrm{the}\:\mathrm{range}\:\mathrm{of}\:\mathrm{the}\:\mathrm{function}\: \\ $$$$\mathrm{f}\left({x}\right)\:=\:\frac{{x}−\mathrm{1}}{\mathrm{p}−{x}^{\mathrm{2}} +\mathrm{1}}\:\mathrm{does}\:\mathrm{not}\:\mathrm{contain}\:\mathrm{any} \\ $$$$\mathrm{values}\:\mathrm{belonging}\:\mathrm{to}\:\mathrm{the}\:\mathrm{interval} \\ $$$$\left[−\mathrm{1},\frac{−\mathrm{1}}{\mathrm{3}}\right]\:{then}\:{true}\:{set}\:\mathrm{of}\:\mathrm{values}\:\mathrm{of}\:\mathrm{p}\:\mathrm{is}\:? \\ $$

Question Number 33990 Answers: 0 Comments: 1

let give I =∫_0 ^1 ln(x)ln(1+x)dx give I at form of serie .

$${let}\:{give}\:{I}\:\:=\int_{\mathrm{0}} ^{\mathrm{1}} {ln}\left({x}\right){ln}\left(\mathrm{1}+{x}\right){dx}\: \\ $$$${give}\:{I}\:{at}\:{form}\:{of}\:{serie}\:. \\ $$

Question Number 33989 Answers: 0 Comments: 0

let f(x)= (e^(−x) /(cosx)) , 2π periodic even developp f at fourier serie .

$${let}\:{f}\left({x}\right)=\:\frac{{e}^{−{x}} }{{cosx}}\:\:\:\:,\:\mathrm{2}\pi\:{periodic}\:{even} \\ $$$${developp}\:{f}\:{at}\:{fourier}\:{serie}\:. \\ $$

Question Number 33988 Answers: 1 Comments: 0

give the algebric form of (1+i)^i .

$${give}\:{the}\:{algebric}\:{form}\:{of}\:\left(\mathrm{1}+{i}\right)^{{i}} . \\ $$

Question Number 33987 Answers: 0 Comments: 2

find ∫_(−∞) ^(+∞) ((cos(αx))/((1+x^2 )^3 )) dx with α≥0 .