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Question Number 90480 Answers: 0 Comments: 0

∫e^x (((1+cos(x))(1−sin(x)))/((e^x cos(x)+1)^2 ))dx

$$\int{e}^{{x}} \frac{\left(\mathrm{1}+{cos}\left({x}\right)\right)\left(\mathrm{1}−{sin}\left({x}\right)\right)}{\left({e}^{{x}} \:{cos}\left({x}\right)+\mathrm{1}\right)^{\mathrm{2}} }{dx} \\ $$

Question Number 90475 Answers: 0 Comments: 3

Question Number 90473 Answers: 1 Comments: 2

Question Number 90472 Answers: 0 Comments: 3

Σ_(k=0) ^m ((2k+3)/2^(m−k) )

$$\underset{\mathrm{k}=\mathrm{0}} {\overset{\mathrm{m}} {\sum}}\frac{\mathrm{2k}+\mathrm{3}}{\mathrm{2}^{\mathrm{m}−\mathrm{k}} } \\ $$

Question Number 90471 Answers: 0 Comments: 0

is there a simple way to write Σ_(n=1) ^∞ ((2/(n(n+1))))^m for any m≥0

$${is}\:{there}\:{a}\:{simple}\:{way} \\ $$$${to}\:{write} \\ $$$$\underset{{n}=\mathrm{1}} {\overset{\infty} {\sum}}\left(\frac{\mathrm{2}}{{n}\left({n}+\mathrm{1}\right)}\right)^{{m}} \:{for}\:{any}\:{m}\geqslant\mathrm{0} \\ $$

Question Number 90470 Answers: 0 Comments: 0

For any positive integer n, τ(n) is the number of its factors Prove, Σ_(i=1) ^n τ(i)=Σ_(i=1) ^n ⌊n/i⌋

$${For}\:{any}\:{positive}\:{integer}\:{n},\:\tau\left({n}\right)\:{is}\:{the}\:{number}\:{of}\:{its}\:{factors}\: \\ $$$${Prove}, \\ $$$$\sum_{{i}=\mathrm{1}} ^{{n}} \tau\left({i}\right)=\sum_{{i}=\mathrm{1}} ^{{n}} \lfloor{n}/{i}\rfloor \\ $$

Question Number 90468 Answers: 0 Comments: 1

Question Number 90499 Answers: 1 Comments: 6

Question Number 90463 Answers: 0 Comments: 1

Question Number 90458 Answers: 0 Comments: 1

∫ (√(x^2 +((13)/x))) dx ?

$$\int\:\sqrt{\mathrm{x}^{\mathrm{2}} +\frac{\mathrm{13}}{\mathrm{x}}}\:\mathrm{dx}\:? \\ $$

Question Number 90457 Answers: 0 Comments: 5

the range of y=(√x) is[0,+∞) if x≥0 or just [0,+∞) ?

$${the}\:{range}\:{of}\:{y}=\sqrt{{x}}\:\:{is}\left[\mathrm{0},+\infty\right)\:{if}\:{x}\geqslant\mathrm{0} \\ $$$${or}\:{just}\:\left[\mathrm{0},+\infty\right)\:? \\ $$

Question Number 90456 Answers: 2 Comments: 0

ABCD is a square with side length=1 E is a moving point between B&C F is a moving point between C&D Find the maximum radius of inscribed circle in △AEF

$${ABCD}\:{is}\:{a}\:{square}\:{with}\:{side}\:{length}=\mathrm{1} \\ $$$${E}\:{is}\:{a}\:{moving}\:{point}\:{between}\:{B\&C} \\ $$$${F}\:{is}\:{a}\:{moving}\:{point}\:{between}\:{C\&D} \\ $$$${Find}\:{the}\:{maximum}\:{radius}\:{of}\:{inscribed} \\ $$$${circle}\:{in}\:\bigtriangleup{AEF} \\ $$

Question Number 90446 Answers: 0 Comments: 2

find the volume of the solid formed by rotating the area trapped by y = sin x and the x−axis around the line y=3 for 0<x<π

$$\mathrm{find}\:\mathrm{the}\:\mathrm{volume}\:\mathrm{of}\:\mathrm{the}\:\mathrm{solid} \\ $$$$\mathrm{formed}\:\mathrm{by}\:\mathrm{rotating}\:\mathrm{the}\:\mathrm{area} \\ $$$$\mathrm{trapped}\:\mathrm{by}\:\mathrm{y}\:=\:\mathrm{sin}\:\mathrm{x}\:\mathrm{and}\:\mathrm{the} \\ $$$$\mathrm{x}−\mathrm{axis}\:\mathrm{around}\:\mathrm{the}\:\mathrm{line}\:\mathrm{y}=\mathrm{3} \\ $$$$\mathrm{for}\:\mathrm{0}<\mathrm{x}<\pi\: \\ $$

Question Number 90443 Answers: 0 Comments: 3

find the volume solid formed by rotating the area trapped between the line y = 1 and the function f(x)=4−3x^2 aroud the line y = 1

$$\mathrm{find}\:\mathrm{the}\:\mathrm{volume}\:\mathrm{solid}\:\mathrm{formed} \\ $$$$\mathrm{by}\:\mathrm{rotating}\:\mathrm{the}\:\mathrm{area}\:\mathrm{trapped} \\ $$$$\mathrm{between}\:\mathrm{the}\:\mathrm{line}\:\mathrm{y}\:=\:\mathrm{1}\:\mathrm{and} \\ $$$$\mathrm{the}\:\mathrm{function}\:\mathrm{f}\left(\mathrm{x}\right)=\mathrm{4}−\mathrm{3x}^{\mathrm{2}} \\ $$$$\mathrm{aroud}\:\mathrm{the}\:\mathrm{line}\:\mathrm{y}\:=\:\mathrm{1} \\ $$

Question Number 90437 Answers: 0 Comments: 2

Question Number 90435 Answers: 0 Comments: 1

Question Number 90434 Answers: 0 Comments: 3

((log_2 (8x).log_3 (27x))/(x^2 −∣x∣)) ≤ 0

$$\frac{\mathrm{log}_{\mathrm{2}} \left(\mathrm{8}{x}\right).\mathrm{log}_{\mathrm{3}} \left(\mathrm{27}{x}\right)}{{x}^{\mathrm{2}} −\mid{x}\mid}\:\leqslant\:\mathrm{0}\: \\ $$

Question Number 90424 Answers: 0 Comments: 1

∫_0 ^2 ∫_((1/2)y) ^1 e^(−x^2 ) dxdy = ?

$$\underset{\mathrm{0}} {\overset{\mathrm{2}} {\int}}\:\underset{\frac{\mathrm{1}}{\mathrm{2}}{y}} {\overset{\mathrm{1}} {\int}}\:{e}^{−{x}^{\mathrm{2}} } \:{dxdy}\:=\:?\: \\ $$

Question Number 90417 Answers: 0 Comments: 6

∫ (dx/(1+x^(12) ))

$$\int\:\frac{\mathrm{dx}}{\mathrm{1}+\mathrm{x}^{\mathrm{12}} } \\ $$

Question Number 90407 Answers: 0 Comments: 0

Σ_(n=0) ^∞ (n^p /(n!)) in terms of p

$$\underset{{n}=\mathrm{0}} {\overset{\infty} {\sum}}\frac{{n}^{{p}} }{{n}!}\:{in}\:{terms}\:{of}\:{p} \\ $$

Question Number 90406 Answers: 2 Comments: 0

If x − (1/x) = 3 x^4 − (1/x^4 ) = ???

$$\mathrm{If}\:\:\:\:\:\:\mathrm{x}\:\:−\:\:\frac{\mathrm{1}}{\mathrm{x}}\:\:\:=\:\:\:\mathrm{3} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\mathrm{x}^{\mathrm{4}} \:\:−\:\:\frac{\mathrm{1}}{\mathrm{x}^{\mathrm{4}} }\:\:\:=\:\:\:??? \\ $$

Question Number 90402 Answers: 1 Comments: 0

Question Number 90432 Answers: 1 Comments: 1

lim_(x→−∞) x[(√(x^2 +1))−x ] =?

$$\underset{{x}\rightarrow−\infty} {\mathrm{lim}}\:\mathrm{x}\left[\sqrt{\mathrm{x}^{\mathrm{2}} +\mathrm{1}}−\mathrm{x}\:\right]\:=? \\ $$

Question Number 90394 Answers: 1 Comments: 2

(3x^2 +9xy+5y^2 )dx = (6x^2 +4xy)dy

$$\left(\mathrm{3x}^{\mathrm{2}} +\mathrm{9xy}+\mathrm{5y}^{\mathrm{2}} \right)\mathrm{dx}\:=\:\left(\mathrm{6x}^{\mathrm{2}} +\mathrm{4xy}\right)\mathrm{dy} \\ $$

Question Number 90383 Answers: 0 Comments: 1

find the values of a and b such that the following function differentiable at x=1 f(x) = { ((x^2 , x≤1)),((2ax+b , x>1)) :}

$$\mathrm{find}\:\mathrm{the}\:\mathrm{values}\:\mathrm{of}\:\mathrm{a}\:\mathrm{and}\:\mathrm{b}\: \\ $$$$\mathrm{such}\:\mathrm{that}\:\mathrm{the}\:\mathrm{following} \\ $$$$\mathrm{function}\:\mathrm{differentiable}\:\mathrm{at}\: \\ $$$$\mathrm{x}=\mathrm{1}\:\mathrm{f}\left(\mathrm{x}\right)\:=\:\begin{cases}{\mathrm{x}^{\mathrm{2}} ,\:\mathrm{x}\leqslant\mathrm{1}}\\{\mathrm{2ax}+\mathrm{b}\:,\:\mathrm{x}>\mathrm{1}}\end{cases} \\ $$

Question Number 90379 Answers: 0 Comments: 2

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