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Question Number 207657 Answers: 0 Comments: 1

Question Number 207652 Answers: 1 Comments: 0

∫_0 ^1 log(1+x^3 )dx = ?and ∫_0 ^1 log (1+x^4 )dx = ? and if possible then find the value of p p = ∫_0 ^1 log(1+x^n )dx = ? n∈N

$$\:\:\:\int_{\mathrm{0}} ^{\mathrm{1}} \mathrm{log}\left(\mathrm{1}+{x}^{\mathrm{3}} \right){dx}\:\:=\:?{and}\:\:\int_{\mathrm{0}} ^{\mathrm{1}} \mathrm{log}\:\left(\mathrm{1}+{x}^{\mathrm{4}} \right){dx}\:=\:? \\ $$$$\:\:\mathrm{and}\:\mathrm{if}\:\mathrm{possible}\:\mathrm{then}\:\mathrm{find}\:\mathrm{the}\:\mathrm{value}\:\mathrm{of}\:\:\mathrm{p} \\ $$$$\:\mathrm{p}\:\:\:=\:\:\:\int_{\mathrm{0}} ^{\mathrm{1}} \mathrm{log}\left(\mathrm{1}+{x}^{{n}} \right){dx}\:=\:?\:\:\:\:\:\:{n}\in\mathbb{N} \\ $$

Question Number 207648 Answers: 1 Comments: 1

f_n (x)=(n/(1+x^2 ))sin((x/n)) /x∈[0,1] , n≥1 calculer lim n→∞∫_0 ^1 f_n (x)dx

$${f}_{{n}} \left({x}\right)=\frac{{n}}{\mathrm{1}+{x}^{\mathrm{2}} }{sin}\left(\frac{{x}}{{n}}\right)\:\:/{x}\in\left[\mathrm{0},\mathrm{1}\right]\:,\:\:{n}\geqslant\mathrm{1} \\ $$$${calculer}\:{lim}\:{n}\rightarrow\infty\int_{\mathrm{0}} ^{\mathrm{1}} {f}_{{n}} \left({x}\right){dx} \\ $$

Question Number 207641 Answers: 1 Comments: 1

Find: 4 cos^2 40 − (1/(cos 20)) = ?

$$\mathrm{Find}:\:\:\:\mathrm{4}\:\mathrm{cos}^{\mathrm{2}} \:\mathrm{40}\:−\:\frac{\mathrm{1}}{\mathrm{cos}\:\mathrm{20}}\:\:=\:\:? \\ $$

Question Number 207639 Answers: 0 Comments: 1

Question Number 207638 Answers: 2 Comments: 4

Question Number 207615 Answers: 2 Comments: 1

{ ((u_(n+1) =((au_n +b)/(cu_n +d)))),((u_0 =k)) :} find u_n in terms of n.

$$\begin{cases}{{u}_{{n}+\mathrm{1}} =\frac{{au}_{{n}} +{b}}{{cu}_{{n}} +{d}}}\\{{u}_{\mathrm{0}} ={k}}\end{cases} \\ $$$${find}\:{u}_{{n}} \:{in}\:{terms}\:{of}\:{n}. \\ $$

Question Number 207611 Answers: 1 Comments: 0

Question Number 207610 Answers: 1 Comments: 1

we have 100 money if we want to buy 100 Donkys Horses and Camels mixed while 20 Donkys cost 1 money, 1 Horse costs 1 money and 1 Camel costs 5 money. find total number of Donkys, Horses, and Camels.

$${we}\:{have}\:\mathrm{100}\:{money}\:{if}\:{we}\:{want}\:{to}\:{buy}\:\mathrm{100}\:{Donkys}\:{Horses} \\ $$$${and}\:{Camels}\:{mixed}\:{while}\:\mathrm{20}\:{Donkys} \\ $$$${cost}\:\mathrm{1}\:{money},\:\mathrm{1}\:{Horse}\:{costs}\:\mathrm{1}\:{money}\:{and} \\ $$$$\mathrm{1}\:{Camel}\:{costs}\:\mathrm{5}\:{money}. \\ $$$${find}\:{total}\:{number}\:{of}\:{Donkys},\:{Horses}, \\ $$$${and}\:{Camels}. \\ $$

Question Number 207597 Answers: 3 Comments: 0

Question Number 207594 Answers: 2 Comments: 0

((xcosθ)/a) + ((ysinθ)/b) = 1 xsinθ − ycosθ = (√(a^2 sin^2 θ + b^2 cos^2 θ)) Eliminate θ.

$$\frac{{x}\mathrm{cos}\theta}{{a}}\:+\:\frac{{y}\mathrm{sin}\theta}{{b}}\:=\:\mathrm{1} \\ $$$${x}\mathrm{sin}\theta\:−\:{y}\mathrm{cos}\theta\:=\:\sqrt{{a}^{\mathrm{2}} \mathrm{sin}^{\mathrm{2}} \theta\:+\:{b}^{\mathrm{2}} \mathrm{cos}^{\mathrm{2}} \theta} \\ $$$$\mathrm{Eliminate}\:\theta. \\ $$

Question Number 207590 Answers: 1 Comments: 1

x^( lg x) < 10^4 The number of roots?

$$\mathrm{x}^{\:\boldsymbol{\mathrm{lg}}\:\boldsymbol{\mathrm{x}}} \:\:<\:\:\mathrm{10}^{\mathrm{4}} \\ $$$$\mathrm{The}\:\mathrm{number}\:\mathrm{of}\:\mathrm{roots}? \\ $$

Question Number 207588 Answers: 1 Comments: 0

(√2) sinx + cosx ≥ 1

$$\sqrt{\mathrm{2}}\:\mathrm{sin}\boldsymbol{\mathrm{x}}\:\:+\:\:\mathrm{cos}\boldsymbol{\mathrm{x}}\:\:\geqslant\:\:\mathrm{1} \\ $$

Question Number 207582 Answers: 2 Comments: 0

∫_0 ^π ln(sinx)dx=−πln2 ∫_0 ^1 lnΓ(x)dx = ln(2π)

$$\int_{\mathrm{0}} ^{\pi} \:{ln}\left({sinx}\right){dx}=−\pi{ln}\mathrm{2} \\ $$$$\int_{\mathrm{0}} ^{\mathrm{1}} {ln}\Gamma\left({x}\right){dx}\:=\:{ln}\left(\mathrm{2}\pi\right) \\ $$

Question Number 207563 Answers: 2 Comments: 0

z + ∣z∣ = 1 + (√3) i find: 𝛟 = ?

$$\mathrm{z}\:\:+\:\:\mid\mathrm{z}\mid\:\:=\:\:\mathrm{1}\:\:+\:\:\sqrt{\mathrm{3}}\:\boldsymbol{\mathrm{i}} \\ $$$$\mathrm{find}:\:\:\:\boldsymbol{\varphi}\:=\:? \\ $$

Question Number 207562 Answers: 0 Comments: 1

4 cos 50° − (1/(sin 20°)) = ?

$$\mathrm{4}\:\mathrm{cos}\:\mathrm{50}°\:−\:\frac{\mathrm{1}}{\mathrm{sin}\:\mathrm{20}°}\:\:=\:\:? \\ $$

Question Number 207561 Answers: 1 Comments: 0

4 sin^2 x + sin 2x = 2 find: x = ?

$$\mathrm{4}\:\mathrm{sin}^{\mathrm{2}} \:\boldsymbol{\mathrm{x}}\:\:+\:\:\mathrm{sin}\:\mathrm{2}\boldsymbol{\mathrm{x}}\:\:=\:\:\mathrm{2} \\ $$$$\mathrm{find}:\:\:\boldsymbol{\mathrm{x}}\:=\:? \\ $$

Question Number 207559 Answers: 1 Comments: 0

log_x (x^2 + 7) ≤ 1

$$\mathrm{log}_{\boldsymbol{\mathrm{x}}} \:\left(\mathrm{x}^{\mathrm{2}} \:+\:\mathrm{7}\right)\:\:\leqslant\:\:\mathrm{1} \\ $$

Question Number 207558 Answers: 1 Comments: 0

(x − 2) lg (x/3) ≥ 0

$$\left(\mathrm{x}\:−\:\mathrm{2}\right)\:\mathrm{lg}\:\frac{\mathrm{x}}{\mathrm{3}}\:\:\geqslant\:\:\mathrm{0} \\ $$

Question Number 207551 Answers: 0 Comments: 0

Question Number 207547 Answers: 1 Comments: 1

lim→+oo ∫_1 ^(+oo) (x^n /(1+x^(n+2) ))dx =?

$${lim}\rightarrow+{oo}\:\int_{\mathrm{1}} ^{+{oo}} \:\frac{{x}^{{n}} }{\mathrm{1}+{x}^{{n}+\mathrm{2}} }{dx}\:=? \\ $$

Question Number 207546 Answers: 1 Comments: 0

The real roots of the equation x^2 +6x+c=0 differ by 2n, where n is a real non−zero. Show that n^2 =9−c Given that the roots also have opposite signs, find the set of possible values of n

$$\mathrm{The}\:\mathrm{real}\:\mathrm{roots}\:\mathrm{of}\:\mathrm{the}\:\mathrm{equation}\:\mathrm{x}^{\mathrm{2}} +\mathrm{6x}+\mathrm{c}=\mathrm{0} \\ $$$$\mathrm{differ}\:\mathrm{by}\:\mathrm{2n},\:\mathrm{where}\:\mathrm{n}\:\mathrm{is}\:\mathrm{a}\:\mathrm{real}\:\mathrm{non}−\mathrm{zero}. \\ $$$$\mathrm{Show}\:\mathrm{that}\:\mathrm{n}^{\mathrm{2}} =\mathrm{9}−\mathrm{c} \\ $$$$\mathrm{Given}\:\mathrm{that}\:\mathrm{the}\:\mathrm{roots}\:\mathrm{also}\:\mathrm{have}\:\mathrm{opposite} \\ $$$$\mathrm{signs},\:\mathrm{find}\:\mathrm{the}\:\mathrm{set}\:\mathrm{of}\:\mathrm{possible}\:\mathrm{values}\:\mathrm{of}\:\mathrm{n} \\ $$

Question Number 207576 Answers: 1 Comments: 2

Question Number 207543 Answers: 2 Comments: 1

Question Number 207565 Answers: 2 Comments: 0

find ∫_0 ^(π/2) (x^2 /(tan^2 x))dx

$${find}\:\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \:\frac{{x}^{\mathrm{2}} }{{tan}^{\mathrm{2}} {x}}{dx} \\ $$

Question Number 207533 Answers: 2 Comments: 1

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