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Question Number 207710    Answers: 0   Comments: 0

Question Number 207707    Answers: 0   Comments: 0

∫_0 ^(+∞) ((sin^2 (x))/(sin^2 (x)+(xcos (x)+sin (x))^2 ))d(x)

$$\underset{\mathrm{0}} {\overset{+\infty} {\int}}\frac{\mathrm{sin}\:^{\mathrm{2}} \left({x}\right)}{\mathrm{sin}\:^{\mathrm{2}} \left({x}\right)+\left({x}\mathrm{cos}\:\left({x}\right)+\mathrm{sin}\:\left({x}\right)\right)^{\mathrm{2}} }{d}\left({x}\right) \\ $$

Question Number 207699    Answers: 1   Comments: 0

(4/(2x−1)) + ((27)/(3x−1)) + ((125)/(5x−1)) = ((144)/(4x−1)) Find: x = ?

$$\frac{\mathrm{4}}{\mathrm{2x}−\mathrm{1}}\:\:+\:\:\frac{\mathrm{27}}{\mathrm{3x}−\mathrm{1}}\:\:+\:\:\frac{\mathrm{125}}{\mathrm{5x}−\mathrm{1}}\:\:=\:\:\frac{\mathrm{144}}{\mathrm{4x}−\mathrm{1}} \\ $$$$\mathrm{Find}:\:\:\boldsymbol{\mathrm{x}}\:=\:? \\ $$

Question Number 207692    Answers: 0   Comments: 1

Question Number 207691    Answers: 3   Comments: 0

Question Number 207688    Answers: 0   Comments: 0

Q207657

$${Q}\mathrm{207657} \\ $$

Question Number 207687    Answers: 1   Comments: 0

Question Number 207673    Answers: 1   Comments: 0

$$\:\:\:\:\underline{ \:} \\ $$$$ \\ $$

Question Number 207683    Answers: 2   Comments: 0

Question Number 207665    Answers: 3   Comments: 0

(1/1) (((20)),(( 0)) ) +(1/2) (((20)),(( 1)) ) +(1/3) (((20)),(( 2)) ) +...+(1/(21)) (((20)),((20)) ) =?

$$\:\:\:\:\frac{\mathrm{1}}{\mathrm{1}}\begin{pmatrix}{\mathrm{20}}\\{\:\:\mathrm{0}}\end{pmatrix}\:+\frac{\mathrm{1}}{\mathrm{2}}\begin{pmatrix}{\mathrm{20}}\\{\:\:\mathrm{1}}\end{pmatrix}\:+\frac{\mathrm{1}}{\mathrm{3}}\begin{pmatrix}{\mathrm{20}}\\{\:\:\mathrm{2}}\end{pmatrix}\:+...+\frac{\mathrm{1}}{\mathrm{21}}\:\begin{pmatrix}{\mathrm{20}}\\{\mathrm{20}}\end{pmatrix}\:=? \\ $$

Question Number 207664    Answers: 1   Comments: 0

If p+q+r = 0 and A=(((p^2 +q^2 +r^2 )^2 )/((pq)^2 +(qr)^2 +(rp)^2 )) B= ((q^2 −pr)/(p^2 +q^2 +r^2 )) . Find A^2 −4B

$$\:\:\:\:\mathrm{If}\:\mathrm{p}+\mathrm{q}+\mathrm{r}\:=\:\mathrm{0}\:\mathrm{and}\: \\ $$$$\:\:\:\:\mathrm{A}=\frac{\left(\mathrm{p}^{\mathrm{2}} +\mathrm{q}^{\mathrm{2}} +\mathrm{r}^{\mathrm{2}} \right)^{\mathrm{2}} }{\left(\mathrm{pq}\right)^{\mathrm{2}} +\left(\mathrm{qr}\right)^{\mathrm{2}} +\left(\mathrm{rp}\right)^{\mathrm{2}} }\: \\ $$$$\:\:\:\mathrm{B}=\:\frac{\mathrm{q}^{\mathrm{2}} −\mathrm{pr}}{\mathrm{p}^{\mathrm{2}} +\mathrm{q}^{\mathrm{2}} +\mathrm{r}^{\mathrm{2}} }\:. \\ $$$$\:\:\mathrm{Find}\:\mathrm{A}^{\mathrm{2}} −\mathrm{4B}\: \\ $$

Question Number 207663    Answers: 0   Comments: 0

Given x_1 +x_3 +...+x_(2023) = 25−(x_2 +x_4 +...+x_(2022) ) x_1 ^(2 ) + x_3 ^2 +...+x_(2023) ^2 = 125−(x_2 ^2 +x_4 ^2 +...+x_(2022) ^2 ) −2≤x_i ≤1 , i=1,2,3,...,2023 x_i integer number Find minimum value of x_1 ^3 +x_3 ^3 +...+x_(2023) ^3

$$\:\:\mathrm{Given}\: \\ $$$$\:\:\:\mathrm{x}_{\mathrm{1}} +\mathrm{x}_{\mathrm{3}} +...+\mathrm{x}_{\mathrm{2023}} \:=\:\mathrm{25}−\left(\mathrm{x}_{\mathrm{2}} +\mathrm{x}_{\mathrm{4}} +...+\mathrm{x}_{\mathrm{2022}} \right) \\ $$$$\:\:\mathrm{x}_{\mathrm{1}} ^{\mathrm{2}\:} +\:\mathrm{x}_{\mathrm{3}} ^{\mathrm{2}} +...+\mathrm{x}_{\mathrm{2023}} ^{\mathrm{2}} \:=\:\mathrm{125}−\left(\mathrm{x}_{\mathrm{2}} ^{\mathrm{2}} +\mathrm{x}_{\mathrm{4}} ^{\mathrm{2}} +...+\mathrm{x}_{\mathrm{2022}} ^{\mathrm{2}} \right) \\ $$$$\:\:−\mathrm{2}\leqslant\mathrm{x}_{\mathrm{i}} \leqslant\mathrm{1}\:,\:\mathrm{i}=\mathrm{1},\mathrm{2},\mathrm{3},...,\mathrm{2023}\: \\ $$$$\:\:\:\mathrm{x}_{\mathrm{i}} \:\mathrm{integer}\:\mathrm{number}\: \\ $$$$\:\mathrm{Find}\:\mathrm{minimum}\:\mathrm{value}\:\mathrm{of}\: \\ $$$$\:\:\mathrm{x}_{\mathrm{1}} ^{\mathrm{3}} +\mathrm{x}_{\mathrm{3}} ^{\mathrm{3}} +...+\mathrm{x}_{\mathrm{2023}} ^{\mathrm{3}} \: \\ $$

Question Number 207661    Answers: 2   Comments: 0

P=1+(1/3)+(1/5)+(1/7)+...+(1/(2023)) Q= (1/(1×2023))+(1/(3×2021))+(1/(5×2019))+...+(1/(2023×1)) (P/Q)=?

$$\:\mathrm{P}=\mathrm{1}+\frac{\mathrm{1}}{\mathrm{3}}+\frac{\mathrm{1}}{\mathrm{5}}+\frac{\mathrm{1}}{\mathrm{7}}+...+\frac{\mathrm{1}}{\mathrm{2023}} \\ $$$$\:\mathrm{Q}=\:\frac{\mathrm{1}}{\mathrm{1}×\mathrm{2023}}+\frac{\mathrm{1}}{\mathrm{3}×\mathrm{2021}}+\frac{\mathrm{1}}{\mathrm{5}×\mathrm{2019}}+...+\frac{\mathrm{1}}{\mathrm{2023}×\mathrm{1}} \\ $$$$\:\:\frac{\mathrm{P}}{\mathrm{Q}}=? \\ $$

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

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

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