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Question Number 155774 Answers: 0 Comments: 0

if x;y>0 then prove that: ((x^2 (1+xy)^2 +x^2 y^4 (1+x)^2 +(1+y)^2 )/(1 + xy + x^2 y + xy^2 )) ≥ 3xy

$$\mathrm{if}\:\:\mathrm{x};\mathrm{y}>\mathrm{0}\:\:\mathrm{then}\:\mathrm{prove}\:\mathrm{that}: \\ $$$$\frac{\mathrm{x}^{\mathrm{2}} \left(\mathrm{1}+\mathrm{xy}\right)^{\mathrm{2}} +\mathrm{x}^{\mathrm{2}} \mathrm{y}^{\mathrm{4}} \left(\mathrm{1}+\mathrm{x}\right)^{\mathrm{2}} +\left(\mathrm{1}+\mathrm{y}\right)^{\mathrm{2}} }{\mathrm{1}\:+\:\mathrm{xy}\:+\:\mathrm{x}^{\mathrm{2}} \mathrm{y}\:+\:\mathrm{xy}^{\mathrm{2}} }\:\geqslant\:\mathrm{3xy} \\ $$

Question Number 155772 Answers: 1 Comments: 0

Question Number 155770 Answers: 1 Comments: 1

Question Number 155759 Answers: 0 Comments: 1

∫_0 ^( 1) ln((sin(x)+cos(x))^2 +1) dx

$$\: \\ $$$$\:\:\int_{\mathrm{0}} ^{\:\mathrm{1}} \:\mathrm{ln}\left(\left(\mathrm{sin}\left({x}\right)+\mathrm{cos}\left({x}\right)\right)^{\mathrm{2}} +\mathrm{1}\right)\:{dx} \\ $$$$\: \\ $$

Question Number 155757 Answers: 0 Comments: 3

2x^2 (dy/dx) − 2x^2 = (x−1) y^2 ; y(1) = 2 □ M

$$\mathrm{2}\boldsymbol{{x}}^{\mathrm{2}} \:\frac{\boldsymbol{{dy}}}{\boldsymbol{{dx}}}\:−\:\mathrm{2}\boldsymbol{{x}}^{\mathrm{2}} \:=\:\left(\boldsymbol{{x}}−\mathrm{1}\right)\:\boldsymbol{{y}}^{\mathrm{2}} \:\:;\:\boldsymbol{{y}}\left(\mathrm{1}\right)\:=\:\mathrm{2}\: \\ $$$$ \\ $$$$\square\:\boldsymbol{{M}}\: \\ $$

Question Number 155745 Answers: 2 Comments: 0

Σ_(k=1) ^n ((k/((4k^2 −1)(2k+3))))=?

$$\:\underset{\mathrm{k}=\mathrm{1}} {\overset{\mathrm{n}} {\sum}}\left(\frac{\mathrm{k}}{\left(\mathrm{4k}^{\mathrm{2}} −\mathrm{1}\right)\left(\mathrm{2k}+\mathrm{3}\right)}\right)=? \\ $$

Question Number 155736 Answers: 0 Comments: 0

Question Number 155735 Answers: 1 Comments: 0

Question Number 155729 Answers: 1 Comments: 0

A={(a,b)∈IR^2 / a^2 +b^2 ≤1} prove that A can′t be written as the cartesian product of two parts of IR.

$$\mathrm{A}=\left\{\left({a},{b}\right)\in\mathrm{IR}^{\mathrm{2}} \:/\:{a}^{\mathrm{2}} +{b}^{\mathrm{2}} \leqslant\mathrm{1}\right\} \\ $$$$\mathrm{prove}\:\mathrm{that}\:\mathrm{A}\:\mathrm{can}'\mathrm{t}\:\mathrm{be}\:\mathrm{written}\:\mathrm{as}\:\mathrm{the}\:\mathrm{cartesian} \\ $$$$\mathrm{product}\:\mathrm{of}\:\mathrm{two}\:\mathrm{parts}\:\mathrm{of}\:\mathrm{IR}. \\ $$

Question Number 155724 Answers: 1 Comments: 0

Find: (1/(sin^2 6^° )) + (1/(sin^2 42°)) + (1/(sin^2 66°)) + (1/(sin^2 78°)) = ?

$$\mathrm{Find}: \\ $$$$\frac{\mathrm{1}}{\mathrm{sin}^{\mathrm{2}} \mathrm{6}^{°} }\:+\:\frac{\mathrm{1}}{\mathrm{sin}^{\mathrm{2}} \mathrm{42}°}\:+\:\frac{\mathrm{1}}{\mathrm{sin}^{\mathrm{2}} \mathrm{66}°}\:+\:\frac{\mathrm{1}}{\mathrm{sin}^{\mathrm{2}} \mathrm{78}°}\:=\:? \\ $$

Question Number 155720 Answers: 3 Comments: 1

Question Number 155776 Answers: 0 Comments: 0

if x∈(0;(π/2)) then prove that: ((2 + (1+cotx)(tan^3 x+cot^3 x))/((1+tanx)(1+cotx))) ≥ (3/2)

$$\mathrm{if}\:\:\:\mathrm{x}\in\left(\mathrm{0};\frac{\pi}{\mathrm{2}}\right)\:\:\mathrm{then}\:\mathrm{prove}\:\mathrm{that}: \\ $$$$\frac{\mathrm{2}\:+\:\left(\mathrm{1}+\mathrm{cot}\boldsymbol{\mathrm{x}}\right)\left(\mathrm{tan}^{\mathrm{3}} \boldsymbol{\mathrm{x}}+\mathrm{cot}^{\mathrm{3}} \boldsymbol{\mathrm{x}}\right)}{\left(\mathrm{1}+\mathrm{tan}\boldsymbol{\mathrm{x}}\right)\left(\mathrm{1}+\mathrm{cot}\boldsymbol{\mathrm{x}}\right)}\:\geqslant\:\frac{\mathrm{3}}{\mathrm{2}} \\ $$

Question Number 155710 Answers: 3 Comments: 1

lim_(x−oo) (1/(n(√n))) Σ_(k=1) ^n E((√(k)))

$$\mathrm{li}\underset{{x}−{oo}} {\mathrm{m}}\:\:\:\frac{\mathrm{1}}{{n}\sqrt{{n}}}\:\:\underset{{k}=\mathrm{1}} {\overset{{n}} {\sum}}{E}\left(\sqrt{\left.{k}\right)}\right. \\ $$$$ \\ $$

Question Number 155701 Answers: 0 Comments: 3

Question Number 155692 Answers: 1 Comments: 5

Question Number 155686 Answers: 2 Comments: 0

∫_0 ^(π/2) (dx/(1+tan x)) =?

$$\:\:\int_{\mathrm{0}} ^{\pi/\mathrm{2}} \frac{{dx}}{\mathrm{1}+\mathrm{tan}\:{x}}\:=? \\ $$

Question Number 155677 Answers: 1 Comments: 1

Question Number 155674 Answers: 0 Comments: 0

monster integral ∫_0 ^( (π/4)) ln^2 (sin(2x)+ cos(3x)) dx

$$\: \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\underline{\mathrm{monster}\:\mathrm{integral}} \\ $$$$\: \\ $$$$\:\:\:\:\int_{\mathrm{0}} ^{\:\frac{\pi}{\mathrm{4}}} \:\mathrm{ln}^{\mathrm{2}} \left(\mathrm{sin}\left(\mathrm{2}{x}\right)+\:\mathrm{cos}\left(\mathrm{3}{x}\right)\right)\:{dx} \\ $$$$\: \\ $$$$\: \\ $$

Question Number 155673 Answers: 0 Comments: 0

Question Number 155667 Answers: 0 Comments: 2

Question Number 155657 Answers: 2 Comments: 1

Question Number 155653 Answers: 0 Comments: 0

if a;b;c≥0 and a+b+c=1 prove that 18 Σ ab + 45 Σ a^2 b ≤ 11

$$\mathrm{if}\:\:\mathrm{a};\mathrm{b};\mathrm{c}\geqslant\mathrm{0}\:\:\mathrm{and}\:\:\mathrm{a}+\mathrm{b}+\mathrm{c}=\mathrm{1}\:\:\mathrm{prove}\:\mathrm{that} \\ $$$$\mathrm{18}\:\Sigma\:\mathrm{ab}\:+\:\mathrm{45}\:\Sigma\:\mathrm{a}^{\mathrm{2}} \mathrm{b}\:\leqslant\:\mathrm{11} \\ $$

Question Number 155650 Answers: 2 Comments: 0

Solve for real numbers: (√((x-a)/(x-b))) + (a/x) = (√((x-b)/(x-a))) + (b/x) a;b∈R and a≠b

$$\mathrm{Solve}\:\mathrm{for}\:\mathrm{real}\:\mathrm{numbers}: \\ $$$$\sqrt{\frac{\mathrm{x}-\mathrm{a}}{\mathrm{x}-\mathrm{b}}}\:+\:\frac{\mathrm{a}}{\mathrm{x}}\:=\:\sqrt{\frac{\mathrm{x}-\mathrm{b}}{\mathrm{x}-\mathrm{a}}}\:+\:\frac{\mathrm{b}}{\mathrm{x}} \\ $$$$\mathrm{a};\mathrm{b}\in\mathbb{R}\:\:\mathrm{and}\:\:\mathrm{a}\neq\mathrm{b} \\ $$

Question Number 155643 Answers: 0 Comments: 1

Question Number 155642 Answers: 0 Comments: 3

Question Number 155639 Answers: 1 Comments: 4

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