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Question Number 145259 Answers: 1 Comments: 0

∫ (((3(√x)+2)^5 )/( (√x))) dx = ?

$$\int\:\frac{\left(\mathrm{3}\sqrt{{x}}+\mathrm{2}\right)^{\mathrm{5}} }{\:\sqrt{{x}}}\:{dx}\:=\:? \\ $$

Question Number 145258 Answers: 0 Comments: 0

Question Number 145256 Answers: 1 Comments: 1

x^x^x =((1/2))^(√2) find x

$${x}^{{x}^{{x}} } =\left(\frac{\mathrm{1}}{\mathrm{2}}\right)^{\sqrt{\mathrm{2}}} \\ $$$$\boldsymbol{\mathrm{find}}\:\boldsymbol{\mathrm{x}} \\ $$

Question Number 145250 Answers: 0 Comments: 3

Question Number 145246 Answers: 1 Comments: 0

prove that a triangle inscribed in a circle of radius r having maximum area is an equilateral triangle with side (√3)r.

$$\mathrm{prove}\:\mathrm{that}\:\mathrm{a}\:\mathrm{triangle}\:\mathrm{inscribed}\: \\ $$$$\mathrm{in}\:\mathrm{a}\:\mathrm{circle}\:\mathrm{of}\:\mathrm{radius}\:\mathrm{r}\:\mathrm{having}\:\mathrm{maximum} \\ $$$$\mathrm{area}\:\mathrm{is}\:\mathrm{an}\:\mathrm{equilateral}\:\mathrm{triangle}\:\mathrm{with} \\ $$$$\mathrm{side}\:\sqrt{\mathrm{3}}\mathrm{r}. \\ $$

Question Number 145240 Answers: 1 Comments: 0

Let a,b,c ≥ 0 and a^2 +b^2 +c^2 = 3. Prove that (1) Σ_(cyc) a^3 +Σ_(cyc) (a+b)^3 ≤ 27 (2) a^3 +b^3 +(b+c)^3 +(c+a)^3 ≥ (1/2)[c^3 +(a+b)^3 ] (3) For a≥b≥c≥0, a^3 +b^3 +(b+c)^3 +(c+a)^3 ≤ 2[c^3 +(a+b)^3 ]

$$\mathrm{Let}\:{a},{b},{c}\:\geqslant\:\mathrm{0}\:\mathrm{and}\:{a}^{\mathrm{2}} +{b}^{\mathrm{2}} +{c}^{\mathrm{2}} \:=\:\mathrm{3}.\:\mathrm{Prove}\:\mathrm{that} \\ $$$$\left(\mathrm{1}\right)\:\:\:\:\:\:\:\:\:\:\:\:\:\underset{{cyc}} {\sum}{a}^{\mathrm{3}} +\underset{{cyc}} {\sum}\left({a}+{b}\right)^{\mathrm{3}} \:\leqslant\:\mathrm{27} \\ $$$$\left(\mathrm{2}\right)\:\:\:\:\:\:\:\:\:\:\:\:\:{a}^{\mathrm{3}} +{b}^{\mathrm{3}} +\left({b}+{c}\right)^{\mathrm{3}} +\left({c}+{a}\right)^{\mathrm{3}} \:\geqslant\:\frac{\mathrm{1}}{\mathrm{2}}\left[{c}^{\mathrm{3}} +\left({a}+{b}\right)^{\mathrm{3}} \right] \\ $$$$\left(\mathrm{3}\right)\:\:\:\:\:\:\:\:\:\:\:\:\:\mathrm{For}\:{a}\geqslant{b}\geqslant{c}\geqslant\mathrm{0},\: \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:{a}^{\mathrm{3}} +{b}^{\mathrm{3}} +\left({b}+{c}\right)^{\mathrm{3}} +\left({c}+{a}\right)^{\mathrm{3}} \:\leqslant\:\mathrm{2}\left[{c}^{\mathrm{3}} +\left({a}+{b}\right)^{\mathrm{3}} \right] \\ $$

Question Number 145238 Answers: 2 Comments: 0

Question Number 145231 Answers: 1 Comments: 3

1<a≤b then find ∫_( a) ^( b) tan^(-1) (((3x)/(1-2x^2 )))dx=?

$$\mathrm{1}<{a}\leqslant{b}\:\:{then}\:{find} \\ $$$$\underset{\:\boldsymbol{{a}}} {\overset{\:\boldsymbol{{b}}} {\int}}\:{tan}^{-\mathrm{1}} \left(\frac{\mathrm{3}{x}}{\mathrm{1}-\mathrm{2}{x}^{\mathrm{2}} }\right){dx}=? \\ $$

Question Number 145229 Answers: 0 Comments: 3

Riddle (clue) 1. I have different types 2. I may be considered natural, whole, positive or negative 3. I am the basic building block of mathematics 4. I am often considered reasonable or rational as well as crazy or irrational 5. I may be terminating or repeating 6. You can locate me on a line bearing my name. who am i?

$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\mathrm{Riddle} \\ $$$$\:\:\:\:\left(\mathrm{clue}\right) \\ $$$$\mathrm{1}.\:\mathrm{I}\:\mathrm{have}\:\mathrm{different}\:\mathrm{types} \\ $$$$\mathrm{2}.\:\mathrm{I}\:\mathrm{may}\:\mathrm{be}\:\mathrm{considered}\:\mathrm{natural},\:\mathrm{whole}, \\ $$$$\mathrm{positive}\:\mathrm{or}\:\mathrm{negative} \\ $$$$\mathrm{3}.\:\mathrm{I}\:\mathrm{am}\:\mathrm{the}\:\mathrm{basic}\:\mathrm{building}\:\mathrm{block}\:\mathrm{of} \\ $$$$\mathrm{mathematics} \\ $$$$\mathrm{4}.\:\mathrm{I}\:\mathrm{am}\:\mathrm{often}\:\mathrm{considered}\:\mathrm{reasonable}\:\mathrm{or} \\ $$$$\:\:\:\:\:\:\:\mathrm{rational}\:\mathrm{as}\:\mathrm{well}\:\mathrm{as}\:\mathrm{crazy}\:\mathrm{or}\:\mathrm{irrational} \\ $$$$\mathrm{5}.\:\mathrm{I}\:\:\mathrm{may}\:\mathrm{be}\:\mathrm{terminating}\:\mathrm{or}\:\mathrm{repeating} \\ $$$$\mathrm{6}.\:\mathrm{You}\:\mathrm{can}\:\mathrm{locate}\:\mathrm{me}\:\mathrm{on}\:\mathrm{a}\:\mathrm{line}\:\mathrm{bearing} \\ $$$$\mathrm{my}\:\mathrm{name}. \\ $$$$\:\:\:\:\:\boldsymbol{\mathrm{who}}\:\boldsymbol{\mathrm{am}}\:\boldsymbol{\mathrm{i}}? \\ $$$$ \\ $$

Question Number 145227 Answers: 1 Comments: 1

Question Number 145224 Answers: 1 Comments: 0

(√(2(√3) + 2)) - (√((√3) - (√2))) = ?

$$\sqrt{\mathrm{2}\sqrt{\mathrm{3}}\:+\:\mathrm{2}}\:-\:\sqrt{\sqrt{\mathrm{3}}\:-\:\sqrt{\mathrm{2}}}\:=\:? \\ $$

Question Number 145208 Answers: 1 Comments: 0

If P : Q = tan 2A : cos A Q : R = cos 2A : sin 2A then P : R =?

$$\:\mathrm{If}\:\mathrm{P}\::\:\mathrm{Q}\:=\:\mathrm{tan}\:\mathrm{2A}\::\:\mathrm{cos}\:\mathrm{A} \\ $$$$\:\:\:\:\:\mathrm{Q}\::\:\mathrm{R}\:=\:\mathrm{cos}\:\mathrm{2A}\::\:\mathrm{sin}\:\mathrm{2A} \\ $$$$\:\:\:\:\:\mathrm{then}\:\mathrm{P}\::\:\mathrm{R}\:=? \\ $$

Question Number 145205 Answers: 2 Comments: 0

lim_(x→0) ((1−cos 2(sin (sin x)))/x^2 )=?

$$\:\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\frac{\mathrm{1}−\mathrm{cos}\:\mathrm{2}\left(\mathrm{sin}\:\left(\mathrm{sin}\:\mathrm{x}\right)\right)}{\mathrm{x}^{\mathrm{2}} }=? \\ $$

Question Number 145202 Answers: 2 Comments: 1

Question Number 145200 Answers: 1 Comments: 0

evaluate:: Σ_(n=0) ^∞ (1/(n!(n^4 +n^2 +1)))=(e/2)

$$\mathrm{evaluate}::\:\:\underset{\mathrm{n}=\mathrm{0}} {\overset{\infty} {\sum}}\frac{\mathrm{1}}{\mathrm{n}!\left(\mathrm{n}^{\mathrm{4}} +\mathrm{n}^{\mathrm{2}} +\mathrm{1}\right)}=\frac{\mathrm{e}}{\mathrm{2}} \\ $$

Question Number 145198 Answers: 1 Comments: 0

Let a≥b≥c≥0 , c^3 +(a+b)^3 ≠0 and a^2 +b^2 +c^2 = 3. Prove that (1/2) ≤ ((a^3 +(b+c)^3 +b^3 +(c+a)^3 )/(c^3 +(a+b)^3 )) ≤ 2 Determine when equality holds.

$$\mathrm{Let}\:{a}\geqslant{b}\geqslant{c}\geqslant\mathrm{0}\:,\:{c}^{\mathrm{3}} +\left({a}+{b}\right)^{\mathrm{3}} \neq\mathrm{0}\:\mathrm{and}\:{a}^{\mathrm{2}} +{b}^{\mathrm{2}} +{c}^{\mathrm{2}} \:=\:\mathrm{3}.\:\mathrm{Prove}\:\mathrm{that} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\frac{\mathrm{1}}{\mathrm{2}}\:\leqslant\:\frac{{a}^{\mathrm{3}} +\left({b}+{c}\right)^{\mathrm{3}} +{b}^{\mathrm{3}} +\left({c}+{a}\right)^{\mathrm{3}} }{{c}^{\mathrm{3}} +\left({a}+{b}\right)^{\mathrm{3}} }\:\leqslant\:\mathrm{2} \\ $$$$\mathrm{Determine}\:\mathrm{when}\:\mathrm{equality}\:\mathrm{holds}. \\ $$

Question Number 145197 Answers: 1 Comments: 1

d/dx of x!=?

$${d}/{dx}\:{of}\:{x}!=? \\ $$

Question Number 145190 Answers: 1 Comments: 0

f(x)+f(1−(1/x))=tan^(−1) x,(x≠0) Find f(x)=?

$$\mathrm{f}\left(\mathrm{x}\right)+\mathrm{f}\left(\mathrm{1}−\frac{\mathrm{1}}{\mathrm{x}}\right)=\mathrm{tan}^{−\mathrm{1}} \mathrm{x},\left(\mathrm{x}\neq\mathrm{0}\right) \\ $$$$\mathrm{Find}\:\mathrm{f}\left(\mathrm{x}\right)=? \\ $$

Question Number 145187 Answers: 0 Comments: 0

((3)^(1/3) + (5)^(1/5) )^(120) How many single units are there at the opening of the binomial

$$\left(\sqrt[{\mathrm{3}}]{\mathrm{3}}\:+\:\sqrt[{\mathrm{5}}]{\mathrm{5}}\right)^{\mathrm{120}} \\ $$$${How}\:{many}\:{single}\:{units}\:{are}\:{there}\:{at} \\ $$$${the}\:{opening}\:{of}\:{the}\:{binomial} \\ $$

Question Number 145186 Answers: 0 Comments: 3

f:R→R f(x-1)+f(x+1)=(√3)∙f(x) ; ∀x∈R find f(x-1)+f(x+5)=?