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Question Number 84859 Answers: 0 Comments: 0

show that lim_(n→∞) ∫_0 ^1 ...∫_0 ^1 (n/(x_1 +x_2 +x_3 +...+x_n ))dx_1 dx_2 ...dx_n =2

$${show}\:{that} \\ $$$$\underset{{n}\rightarrow\infty} {{lim}}\int_{\mathrm{0}} ^{\mathrm{1}} ...\int_{\mathrm{0}} ^{\mathrm{1}} \frac{{n}}{{x}_{\mathrm{1}} +{x}_{\mathrm{2}} +{x}_{\mathrm{3}} +...+{x}_{{n}} }{dx}_{\mathrm{1}} {dx}_{\mathrm{2}} ...{dx}_{{n}} =\mathrm{2}\: \\ $$

Question Number 84849 Answers: 0 Comments: 5

ABC is a triangle prove that sinA+sinB+sinC>sinA sinB sinC

$${ABC}\:{is}\:{a}\:{triangle}\: \\ $$$${prove}\:{that} \\ $$$${sinA}+{sinB}+{sinC}>{sinA}\:{sinB}\:{sinC} \\ $$

Question Number 84845 Answers: 1 Comments: 1

Question Number 84843 Answers: 0 Comments: 2

∫((sin(7x))/(cos(3x))) dx

$$\int\frac{{sin}\left(\mathrm{7}{x}\right)}{{cos}\left(\mathrm{3}{x}\right)}\:{dx} \\ $$

Question Number 84835 Answers: 0 Comments: 1

Question Number 84834 Answers: 1 Comments: 1

lim_(x→0) ((√(x tan x))/(sin 3x))

$$\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\frac{\sqrt{\mathrm{x}\:\mathrm{tan}\:\mathrm{x}}}{\mathrm{sin}\:\mathrm{3x}} \\ $$

Question Number 84830 Answers: 1 Comments: 0

(dy/dx) = ((1−x−y)/(x+y))

$$\frac{\mathrm{dy}}{\mathrm{dx}}\:=\:\frac{\mathrm{1}−\mathrm{x}−\mathrm{y}}{\mathrm{x}+\mathrm{y}} \\ $$

Question Number 84828 Answers: 0 Comments: 1

log_3 (25x^2 −4)−log_3 (x) ≤ log_3 (26x^2 +((17)/x)−10)

$$\mathrm{log}_{\mathrm{3}} \left(\mathrm{25x}^{\mathrm{2}} −\mathrm{4}\right)−\mathrm{log}_{\mathrm{3}} \left(\mathrm{x}\right)\:\leqslant\:\mathrm{log}_{\mathrm{3}} \left(\mathrm{26x}^{\mathrm{2}} +\frac{\mathrm{17}}{\mathrm{x}}−\mathrm{10}\right) \\ $$

Question Number 84826 Answers: 0 Comments: 1

(√(x^2 −2x+2)) + log_3 (√(x^2 −2x+10)) = 2

$$\sqrt{\mathrm{x}^{\mathrm{2}} −\mathrm{2x}+\mathrm{2}}\:+\:\mathrm{log}_{\mathrm{3}} \:\sqrt{\mathrm{x}^{\mathrm{2}} −\mathrm{2x}+\mathrm{10}}\:=\:\mathrm{2} \\ $$

Question Number 84820 Answers: 1 Comments: 0

Question Number 84814 Answers: 0 Comments: 1

1.Finx

$$\mathrm{1}.{Finx} \\ $$

Question Number 84810 Answers: 0 Comments: 1

∫_0 ^π ln(((1+b cos(x))/(1+a sin(x)))) dx −1<a<b<1

$$\int_{\mathrm{0}} ^{\pi} {ln}\left(\frac{\mathrm{1}+{b}\:{cos}\left({x}\right)}{\mathrm{1}+{a}\:{sin}\left({x}\right)}\right)\:{dx} \\ $$$$−\mathrm{1}<{a}<{b}<\mathrm{1} \\ $$

Question Number 84809 Answers: 1 Comments: 1

∫(x/((x^2 +1)^(3/2) arctan(x))) dx

$$\int\frac{{x}}{\left({x}^{\mathrm{2}} +\mathrm{1}\right)^{\frac{\mathrm{3}}{\mathrm{2}}} {arctan}\left({x}\right)}\:{dx} \\ $$

Question Number 84861 Answers: 0 Comments: 1

If f(x) is an even function, then ∫_( 0) ^π f (cos x) dx = 2∫_( 0) ^(π/2) f (cos x) dx

$$\mathrm{If}\:{f}\left({x}\right)\:\mathrm{is}\:\mathrm{an}\:\mathrm{even}\:\mathrm{function},\:\mathrm{then} \\ $$$$\underset{\:\mathrm{0}} {\overset{\pi} {\int}}\:{f}\:\left(\mathrm{cos}\:{x}\right)\:{dx}\:=\:\mathrm{2}\underset{\:\mathrm{0}} {\overset{\pi/\mathrm{2}} {\int}}\:{f}\:\left(\mathrm{cos}\:{x}\right)\:{dx} \\ $$

Question Number 84792 Answers: 0 Comments: 0

a,b,c≥0 a+b+c=3 show that (a)^(1/3) +(b)^(1/3) +(c)^(1/3) ≥ab+bc+ca

$${a},{b},{c}\geqslant\mathrm{0} \\ $$$${a}+{b}+{c}=\mathrm{3} \\ $$$${show}\:{that} \\ $$$$\sqrt[{\mathrm{3}}]{{a}}\:+\sqrt[{\mathrm{3}}]{{b}}\:+\sqrt[{\mathrm{3}}]{{c}}\geqslant{ab}+{bc}+{ca} \\ $$

Question Number 84783 Answers: 1 Comments: 6

Question Number 84782 Answers: 1 Comments: 8

Find the last three digits of 2019^(2019) .

$${Find}\:{the}\:{last}\:{three}\:{digits}\:{of}\:\mathrm{2019}^{\mathrm{2019}} . \\ $$

Question Number 84778 Answers: 0 Comments: 0

let f(x)=(1/(2+sinx)) developp f at fourier serie

$${let}\:{f}\left({x}\right)=\frac{\mathrm{1}}{\mathrm{2}+{sinx}} \\ $$$${developp}\:{f}\:{at}\:{fourier}\:{serie} \\ $$

Question Number 84770 Answers: 2 Comments: 1

x^2 +y^2 = 30 (1/x)+(1/y) = 2 find the solution x & y ?

$$\mathrm{x}^{\mathrm{2}} +\mathrm{y}^{\mathrm{2}} \:=\:\mathrm{30}\: \\ $$$$\frac{\mathrm{1}}{\mathrm{x}}+\frac{\mathrm{1}}{\mathrm{y}}\:=\:\mathrm{2}\: \\ $$$$\mathrm{find}\:\mathrm{the}\:\mathrm{solution}\:\mathrm{x}\:\&\:\mathrm{y}\:? \\ $$

Question Number 84767 Answers: 1 Comments: 0

Find all solutions of 2020x^2 − y^2 = 6059 x, y < 2020 , x, y ∈ N

$${Find}\:\:{all}\:\:{solutions}\:\:{of} \\ $$$$\:\:\:\:\:\:\:\mathrm{2020}{x}^{\mathrm{2}} \:−\:{y}^{\mathrm{2}} \:\:=\:\:\mathrm{6059} \\ $$$${x},\:{y}\:<\:\:\mathrm{2020}\:\:,\:\:\:{x},\:{y}\:\:\in\:\:\mathbb{N} \\ $$

Question Number 84766 Answers: 1 Comments: 2

∫ (dx/((16+9sin x)^2 ))

$$\int\:\frac{\mathrm{dx}}{\left(\mathrm{16}+\mathrm{9sin}\:\mathrm{x}\right)^{\mathrm{2}} } \\ $$$$ \\ $$

Question Number 84759 Answers: 1 Comments: 0

calculate ∫_(−∞) ^(+∞) ((arctan(2x^2 ))/(1+x^2 ))dx

$${calculate}\:\:\int_{−\infty} ^{+\infty} \:\:\frac{{arctan}\left(\mathrm{2}{x}^{\mathrm{2}} \right)}{\mathrm{1}+{x}^{\mathrm{2}} }{dx} \\ $$

Question Number 84740 Answers: 1 Comments: 5

find the remainder when −18 is divided by 4

$$\mathrm{find}\:\mathrm{the}\:\mathrm{remainder}\:\mathrm{when}\:−\mathrm{18}\:\mathrm{is}\:\mathrm{divided}\:\mathrm{by}\:\mathrm{4} \\ $$

Question Number 84739 Answers: 1 Comments: 0

find the unit digit in the number 15^(1789) + 17^(1789) + 19^(1789)

$$\mathrm{find}\:\mathrm{the}\:\mathrm{unit}\:\mathrm{digit}\:\mathrm{in}\:\mathrm{the}\:\mathrm{number} \\ $$$$\:\mathrm{15}^{\mathrm{1789}} \:+\:\mathrm{17}^{\mathrm{1789}} \:+\:\mathrm{19}^{\mathrm{1789}} \\ $$

Question Number 84733 Answers: 1 Comments: 0

∫((x^4 +1)/(x^6 +1))dx

$$\int\frac{{x}^{\mathrm{4}} +\mathrm{1}}{{x}^{\mathrm{6}} +\mathrm{1}}{dx} \\ $$

Question Number 84728 Answers: 0 Comments: 8

y=x[x[x]] with x∈R^+ find the range of function and solve x[x[x]]=150.

$${y}={x}\left[{x}\left[{x}\right]\right]\:{with}\:{x}\in{R}^{+} \\ $$$${find}\:{the}\:{range}\:{of}\:{function} \\ $$$${and}\:{solve}\:{x}\left[{x}\left[{x}\right]\right]=\mathrm{150}. \\ $$

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