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Question Number 145319 Answers: 1 Comments: 0

# Calculus# Σ_(n=0) ^∞ (1/(n! + (n + 1 )!)) =?

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

Question Number 145318 Answers: 1 Comments: 0

Let f:[0,1]→R be a differentiable function such that f(f(x))=x for all x∈[0,1] and f(0)=1. If n is a positive integer, evaluate the following integral: ∫_0 ^( 1) (x−f(x))^(2n) dx

$$\mathrm{Let}\:{f}:\left[\mathrm{0},\mathrm{1}\right]\rightarrow\mathbb{R}\:\mathrm{be}\:\mathrm{a}\:\mathrm{differentiable}\:\mathrm{function} \\ $$$$\mathrm{such}\:\mathrm{that}\:{f}\left({f}\left({x}\right)\right)={x}\:\mathrm{for}\:\mathrm{all}\:{x}\in\left[\mathrm{0},\mathrm{1}\right]\:\mathrm{and} \\ $$$${f}\left(\mathrm{0}\right)=\mathrm{1}. \\ $$$$\mathrm{If}\:{n}\:\mathrm{is}\:\mathrm{a}\:\mathrm{positive}\:\mathrm{integer},\:\mathrm{evaluate}\:\mathrm{the} \\ $$$$\mathrm{following}\:\mathrm{integral}:\: \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\int_{\mathrm{0}} ^{\:\mathrm{1}} \left({x}−{f}\left({x}\right)\right)^{\mathrm{2}{n}} \:{dx} \\ $$

Question Number 145316 Answers: 2 Comments: 2

Question Number 145314 Answers: 0 Comments: 2

Let a,b ≥ 0 and (a+1)(b+1) = (a+b)^2 . Prove that (a+b)(√((a+1)^3 +(b+1)^3 )) ≤ (a+1)^2 +(b+1)^2 ≤ (1/2)[(a+1)^3 +(b+1)^3 ]

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

Question Number 145309 Answers: 0 Comments: 0

Question Number 145304 Answers: 1 Comments: 0

1+((3x)/(1!)) +((5x^2 )/(2!))+((7x^3 )/(3!))+((9x^4 )/(4!))+...+∞=?

$$\:\mathrm{1}+\frac{\mathrm{3x}}{\mathrm{1}!}\:+\frac{\mathrm{5x}^{\mathrm{2}} }{\mathrm{2}!}+\frac{\mathrm{7x}^{\mathrm{3}} }{\mathrm{3}!}+\frac{\mathrm{9x}^{\mathrm{4}} }{\mathrm{4}!}+...+\infty=? \\ $$

Question Number 145305 Answers: 1 Comments: 2

Re^ soudre (((8/(sin^2 (x))) + 1)/((1/(cos^2 (x))) + tan^2 (x))) = cotan^2 (x)+(4/3)

$$\mathrm{R}\acute {\mathrm{e}soudre}\:\:\:\frac{\frac{\mathrm{8}}{\mathrm{sin}^{\mathrm{2}} \left(\mathrm{x}\right)}\:+\:\mathrm{1}}{\frac{\mathrm{1}}{\mathrm{cos}^{\mathrm{2}} \left(\mathrm{x}\right)}\:+\:\mathrm{tan}^{\mathrm{2}} \left(\mathrm{x}\right)}\:=\:\mathrm{cotan}^{\mathrm{2}} \left(\mathrm{x}\right)+\frac{\mathrm{4}}{\mathrm{3}} \\ $$

Question Number 145300 Answers: 1 Comments: 0

yy′ = x e^(x/y)

$$\:\:\:\:\:\:\:\mathrm{yy}'\:=\:\mathrm{x}\:\mathrm{e}^{\frac{\mathrm{x}}{\mathrm{y}}} \: \\ $$

Question Number 145294 Answers: 2 Comments: 0

Given a polynomial p(x)=x^4 +4x^3 +(2p+2)x^2 +(2p+5q+2)x+3q+2r. If p(x)= (x^3 +2x^2 +8x+6)Q(x) then what the value of (p+2q)r .

$$\mathrm{Given}\:\mathrm{a}\:\mathrm{polynomial}\: \\ $$$$\mathrm{p}\left(\mathrm{x}\right)=\mathrm{x}^{\mathrm{4}} +\mathrm{4x}^{\mathrm{3}} +\left(\mathrm{2p}+\mathrm{2}\right)\mathrm{x}^{\mathrm{2}} +\left(\mathrm{2p}+\mathrm{5q}+\mathrm{2}\right)\mathrm{x}+\mathrm{3q}+\mathrm{2r}. \\ $$$$\mathrm{If}\:\mathrm{p}\left(\mathrm{x}\right)=\:\left(\mathrm{x}^{\mathrm{3}} +\mathrm{2x}^{\mathrm{2}} +\mathrm{8x}+\mathrm{6}\right)\mathrm{Q}\left(\mathrm{x}\right) \\ $$$$\:\mathrm{then}\:\mathrm{what}\:\mathrm{the}\:\mathrm{value}\:\mathrm{of}\: \\ $$$$\:\left(\mathrm{p}+\mathrm{2q}\right)\mathrm{r}\:. \\ $$

Question Number 145290 Answers: 2 Comments: 0

Question Number 145286 Answers: 3 Comments: 0

Given the function f(x) =((6x^2 −x^3 ))^(1/3) Find the oblique assymptote(s) of the function.

$$\mathrm{Given}\:\mathrm{the}\:\mathrm{function} \\ $$$$\:{f}\left({x}\right)\:=\sqrt[{\mathrm{3}}]{\mathrm{6}{x}^{\mathrm{2}} −{x}^{\mathrm{3}} } \\ $$$$\mathrm{Find}\:\mathrm{the}\:\mathrm{oblique}\:\mathrm{assymptote}\left(\mathrm{s}\right)\:\mathrm{of}\:\mathrm{the}\:\mathrm{function}. \\ $$

Question Number 145285 Answers: 0 Comments: 1

x=2−2+2−2+2−…+2−2+2=2

$${x}=\mathrm{2}−\mathrm{2}+\mathrm{2}−\mathrm{2}+\mathrm{2}−\ldots+\mathrm{2}−\mathrm{2}+\mathrm{2}=\mathrm{2} \\ $$

Question Number 145282 Answers: 1 Comments: 0

Question Number 145280 Answers: 1 Comments: 0

if x^2 =x+2 find ((x^3 +1)/((x+1)^2 )) = ?

$${if}\:\:\boldsymbol{{x}}^{\mathrm{2}} =\boldsymbol{{x}}+\mathrm{2}\:\:{find}\:\:\frac{\boldsymbol{{x}}^{\mathrm{3}} +\mathrm{1}}{\left(\boldsymbol{{x}}+\mathrm{1}\right)^{\mathrm{2}} }\:=\:? \\ $$

Question Number 145279 Answers: 1 Comments: 0

2^(a!) + 2^(b!) + 2^(c!) = x Find natural numbers a;b;c such that the number “x” is a cube of any number.

$$\mathrm{2}^{\boldsymbol{{a}}!} \:+\:\mathrm{2}^{\boldsymbol{{b}}!} \:+\:\mathrm{2}^{\boldsymbol{{c}}!} \:=\:\boldsymbol{{x}} \\ $$$${Find}\:{natural}\:{numbers}\:\boldsymbol{{a}};\boldsymbol{{b}};\boldsymbol{{c}}\:{such} \\ $$$${that}\:{the}\:{number}\:``\boldsymbol{{x}}''\:{is}\:{a}\:{cube}\:{of} \\ $$$${any}\:{number}. \\ $$

Question Number 145274 Answers: 1 Comments: 0

∫_(∣z∣=1) ((f^− (z))/(z−a))dz

$$\int_{\mid{z}\mid=\mathrm{1}} \frac{\overset{−} {{f}}\left({z}\right)}{{z}−{a}}{dz} \\ $$

Question Number 145273 Answers: 1 Comments: 0

show that ∀x∈R x−1≤E(x)≤x

$${show}\:{that}\:\forall{x}\in\mathbb{R}\:{x}−\mathrm{1}\leqslant{E}\left({x}\right)\leqslant{x} \\ $$

Question Number 145270 Answers: 1 Comments: 0

# Calculus ( I ) # Σ_(n=1) ^∞ Arccot(3 +((n ( n + 1))/3) )= ? .....

$$ \\ $$$$\:\:\:\:\:\:\:#\:\mathrm{Calculus}\:\left(\:\mathrm{I}\:\right)\:# \\ $$$$\:\:\:\:\:\:\:\underset{{n}=\mathrm{1}} {\overset{\infty} {\sum}}\:\mathrm{Arccot}\left(\mathrm{3}\:+\frac{{n}\:\left(\:{n}\:+\:\mathrm{1}\right)}{\mathrm{3}}\:\right)=\:? \\ $$$$\:\:\:\:\:\:..... \\ $$

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=?

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

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