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Question Number 145359 Answers: 1 Comments: 0

How many digits will there be in 875^(16) ?

$$\mathrm{How}\:\mathrm{many}\:\mathrm{digits}\:\mathrm{will}\:\mathrm{there}\:\mathrm{be} \\ $$$$\mathrm{in}\:\mathrm{875}^{\mathrm{16}} \:? \\ $$

Question Number 145358 Answers: 1 Comments: 0

Evaluate:: ∫_0 ^1 ln(1+x^2 )∙arctan(x)dx=?

$$\mathrm{Evaluate}::\:\:\:\int_{\mathrm{0}} ^{\mathrm{1}} \mathrm{ln}\left(\mathrm{1}+\mathrm{x}^{\mathrm{2}} \right)\centerdot\mathrm{arctan}\left(\mathrm{x}\right)\mathrm{dx}=? \\ $$

Question Number 145345 Answers: 1 Comments: 0

Σ_(n=1) ^∞ (((−1)^n n)/((2n+1)!))=?

$$\underset{\mathrm{n}=\mathrm{1}} {\overset{\infty} {\sum}}\frac{\left(−\mathrm{1}\right)^{\mathrm{n}} \mathrm{n}}{\left(\mathrm{2n}+\mathrm{1}\right)!}=? \\ $$

Question Number 145344 Answers: 2 Comments: 0

z^4 = 49 - 20(√6) ⇒ z=?

$$\boldsymbol{{z}}^{\mathrm{4}} \:=\:\mathrm{49}\:-\:\mathrm{20}\sqrt{\mathrm{6}}\:\:\Rightarrow\:\boldsymbol{{z}}=? \\ $$

Question Number 145339 Answers: 1 Comments: 0

Let f(x)=e^x cos x,Find Σ_(n=0) ^∞ ((f^((n)) (x))/2^n )=?

$$\mathrm{Let}\:\mathrm{f}\left(\mathrm{x}\right)=\mathrm{e}^{\mathrm{x}} \mathrm{cos}\:\mathrm{x},\mathrm{Find}\:\underset{\mathrm{n}=\mathrm{0}} {\overset{\infty} {\sum}}\frac{\mathrm{f}^{\left(\mathrm{n}\right)} \left(\mathrm{x}\right)}{\mathrm{2}^{\mathrm{n}} }=? \\ $$

Question Number 145338 Answers: 1 Comments: 0

de^ rive^ e n-ie^ me de (x^3 /(1+x^6 ))

$$\mathrm{d}\acute {\mathrm{e}riv}\acute {\mathrm{e}e}\:\:\:\mathrm{n}-\mathrm{i}\grave {\mathrm{e}me}\:\:\:\mathrm{de}\:\:\:\frac{\mathrm{x}^{\mathrm{3}} }{\mathrm{1}+\mathrm{x}^{\mathrm{6}} } \\ $$

Question Number 145337 Answers: 1 Comments: 0

The Area of a square,A(t),is increased at a rate equal to its perimeter ,A(t) satisfies the differential equation (dA/dt)= A. 4A B. 2A C.−4(√A) D. 4(√A)

$$\mathrm{The}\:\mathrm{Area}\:\mathrm{of}\:\mathrm{a}\:\mathrm{square},{A}\left({t}\right),\mathrm{is}\:\mathrm{increased}\:\mathrm{at}\:\mathrm{a}\:\mathrm{rate} \\ $$$$\mathrm{equal}\:\mathrm{to}\:\mathrm{its}\:\mathrm{perimeter}\:,{A}\left({t}\right)\:\mathrm{satisfies}\:\mathrm{the}\:\mathrm{differential} \\ $$$$\mathrm{equation}\:\frac{{dA}}{{dt}}= \\ $$$$\mathrm{A}.\:\mathrm{4}{A}\:\:\:\:\:\:\:\:\:\mathrm{B}.\:\mathrm{2}{A}\:\:\:\:\:\:\:\:\:\mathrm{C}.−\mathrm{4}\sqrt{{A}}\:\:\:\:\:\:\:\:\mathrm{D}.\:\mathrm{4}\sqrt{{A}} \\ $$

Question Number 145336 Answers: 0 Comments: 4

Which of the following linear diophantine equations has positive solutions, A. x+ 5y = 7 B. x+5y = 3 B. x + 5y = 2 C. x+ 5y = 1

$$\mathrm{Which}\:\mathrm{of}\:\mathrm{the}\:\mathrm{following}\:\mathrm{linear}\:\mathrm{diophantine}\:\mathrm{equations}\:\mathrm{has} \\ $$$$\mathrm{positive}\:\mathrm{solutions}, \\ $$$$\mathrm{A}.\:{x}+\:\mathrm{5}{y}\:=\:\mathrm{7} \\ $$$$\mathrm{B}.\:{x}+\mathrm{5}{y}\:=\:\mathrm{3} \\ $$$$\mathrm{B}.\:{x}\:+\:\mathrm{5}{y}\:=\:\mathrm{2} \\ $$$$\mathrm{C}.\:{x}+\:\mathrm{5}{y}\:=\:\mathrm{1} \\ $$

Question Number 145324 Answers: 2 Comments: 0

∫ (dx/(1 + x^6 )) = ?

$$\int\:\frac{{dx}}{\mathrm{1}\:+\:{x}^{\mathrm{6}} }\:=\:? \\ $$

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

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

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