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Question Number 89593 Answers: 0 Comments: 5

show that ∫_0 ^(π/2) ln(sec(x)) ln(csc(x)) dx=((π^2 ln^2 (2))/2)−(π^4 /(48))

$${show}\:{that} \\ $$$$\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} {ln}\left({sec}\left({x}\right)\right)\:{ln}\left({csc}\left({x}\right)\right)\:{dx}=\frac{\pi^{\mathrm{2}} \:{ln}^{\mathrm{2}} \left(\mathrm{2}\right)}{\mathrm{2}}−\frac{\pi^{\mathrm{4}} }{\mathrm{48}} \\ $$

Question Number 89425 Answers: 0 Comments: 0

∫_1 ^4 (√(1+((y^3 /2)−(1/2)y^(−1) )^2 )) dy

$$\int_{\mathrm{1}} ^{\mathrm{4}} \sqrt{\mathrm{1}+\left(\frac{{y}^{\mathrm{3}} }{\mathrm{2}}−\frac{\mathrm{1}}{\mathrm{2}}{y}^{−\mathrm{1}} \right)^{\mathrm{2}} }\:{dy} \\ $$$$ \\ $$

Question Number 89422 Answers: 1 Comments: 1

Question Number 89415 Answers: 1 Comments: 4

Question Number 89412 Answers: 0 Comments: 8

Question Number 89399 Answers: 0 Comments: 11

Hi. A ballot box contains 3 red balls, 4 blues balls and 5 white balls. we draw successively 3 balls in ballot box by re−puting the drawn balls. 1)Calculate the number of draws containing one ball of each color.

$$\mathrm{Hi}. \\ $$$$\mathrm{A}\:\mathrm{ballot}\:\mathrm{box}\:\mathrm{contains}\:\mathrm{3}\:\mathrm{red}\:\mathrm{balls},\:\mathrm{4}\:\mathrm{blues} \\ $$$$\mathrm{balls}\:\mathrm{and}\:\mathrm{5}\:\mathrm{white}\:\mathrm{balls}. \\ $$$$\mathrm{we}\:\mathrm{draw}\:\mathrm{successively}\:\mathrm{3}\:\mathrm{balls}\:\mathrm{in}\:\mathrm{ballot}\:\mathrm{box}\: \\ $$$$\mathrm{by}\:\mathrm{re}−\mathrm{puting}\:\mathrm{the}\:\mathrm{drawn}\:\mathrm{balls}. \\ $$$$\left.\mathrm{1}\right)\mathrm{Calculate}\:\mathrm{the}\:\mathrm{number}\:\mathrm{of}\:\mathrm{draws}\: \\ $$$$\mathrm{containing}\:\mathrm{one}\:\mathrm{ball}\:\mathrm{of}\:\mathrm{each}\:\mathrm{color}. \\ $$

Question Number 89385 Answers: 1 Comments: 1

Question Number 89384 Answers: 0 Comments: 3

Question Number 141569 Answers: 0 Comments: 0

lim_(n→+∝) (((110^2 )/((11−10)(11^2 −10^2 ))) + ((110^2 )/((11^2 −10^2 )(11^3 −10^3 ))) + ....+ ((110^2 )/((11^n −10^n )(11^(n+1) −10^(n+1) ))))

$$\underset{\mathrm{n}\rightarrow+\propto} {\mathrm{lim}}\left(\frac{\mathrm{110}^{\mathrm{2}} }{\left(\mathrm{11}−\mathrm{10}\right)\left(\mathrm{11}^{\mathrm{2}} −\mathrm{10}^{\mathrm{2}} \right)}\:+\:\frac{\mathrm{110}^{\mathrm{2}} }{\left(\mathrm{11}^{\mathrm{2}} −\mathrm{10}^{\mathrm{2}} \right)\left(\mathrm{11}^{\mathrm{3}} −\mathrm{10}^{\mathrm{3}} \right)}\:+\:....+\:\frac{\mathrm{110}^{\mathrm{2}} }{\left(\mathrm{11}^{\mathrm{n}} −\mathrm{10}^{\mathrm{n}} \right)\left(\mathrm{11}^{\mathrm{n}+\mathrm{1}} −\mathrm{10}^{\mathrm{n}+\mathrm{1}} \right)}\right) \\ $$

Question Number 89382 Answers: 1 Comments: 2

∫_0 ^∞ e^(−2x) cos(3x) sin(4x)

$$\int_{\mathrm{0}} ^{\infty} {e}^{−\mathrm{2}{x}} \:{cos}\left(\mathrm{3}{x}\right)\:{sin}\left(\mathrm{4}{x}\right) \\ $$

Question Number 89361 Answers: 1 Comments: 0

Question Number 89351 Answers: 0 Comments: 4

prove tan 3^o ×tan 39^o ×tan 89^o = tan 15^o

$${prove}\:\mathrm{tan}\:\mathrm{3}^{{o}} ×\mathrm{tan}\:\mathrm{39}^{{o}} ×\mathrm{tan}\:\mathrm{89}^{{o}} \:=\:\mathrm{tan}\:\mathrm{15}^{{o}} \\ $$

Question Number 89511 Answers: 1 Comments: 6

∫_0 ^(3π) (√(1+sin^4 (θ/3) cos^2 (θ/3))) dθ

$$\int_{\mathrm{0}} ^{\mathrm{3}\pi} \sqrt{\mathrm{1}+{sin}^{\mathrm{4}} \frac{\theta}{\mathrm{3}}\:{cos}^{\mathrm{2}} \frac{\theta}{\mathrm{3}}}\:{d}\theta \\ $$

Question Number 89510 Answers: 0 Comments: 3

Find the expansion of Xe^(1/x)

$${Find}\:{the}\:{expansion}\:{of}\:{Xe}^{\frac{\mathrm{1}}{{x}}} \: \\ $$

Question Number 89344 Answers: 0 Comments: 6

(2x−1)^2 +8(√(2xy)) = 4 4y−(√(8xy−1)) = 1

$$\left(\mathrm{2}{x}−\mathrm{1}\right)^{\mathrm{2}} +\mathrm{8}\sqrt{\mathrm{2}{xy}}\:=\:\mathrm{4} \\ $$$$\mathrm{4}{y}−\sqrt{\mathrm{8}{xy}−\mathrm{1}}\:=\:\mathrm{1} \\ $$

Question Number 89343 Answers: 0 Comments: 2

Given α and β ∈N such that I(α,β)=∫_ ^1 t^α (1−t)^β dt show that I(α;β)=((α!β!)/((α+β+1)!))

$${Given}\:\alpha\:{and}\:\beta\:\in\mathbb{N}\:{such}\:{that} \\ $$$${I}\left(\alpha,\beta\right)=\int_{} ^{\mathrm{1}} {t}^{\alpha} \left(\mathrm{1}−{t}\right)^{\beta} {dt} \\ $$$${show}\:{that} \\ $$$${I}\left(\alpha;\beta\right)=\frac{\alpha!\beta!}{\left(\alpha+\beta+\mathrm{1}\right)!} \\ $$

Question Number 89341 Answers: 0 Comments: 3

lim_(n→∞) ((Π_(k=1) ^(n−1) sin(((kπ)/(2n)))))^(1/n) =?

$$\underset{{n}\rightarrow\infty} {{lim}}\sqrt[{{n}}]{\underset{{k}=\mathrm{1}} {\overset{{n}−\mathrm{1}} {\prod}}{sin}\left(\frac{{k}\pi}{\mathrm{2}{n}}\right)}=? \\ $$

Question Number 89327 Answers: 1 Comments: 2

Question Number 89324 Answers: 0 Comments: 1

Question Number 89322 Answers: 0 Comments: 2

∫ (x^2 /(1 + 5^x )) dx

$$\int\:\frac{\mathrm{x}^{\mathrm{2}} }{\mathrm{1}\:+\:\mathrm{5}^{\mathrm{x}} }\:\mathrm{dx} \\ $$

Question Number 89320 Answers: 0 Comments: 1

Question Number 89319 Answers: 1 Comments: 0

A ball is projected from a point O with an initial velocity u and angle θ with the horizontal ground. Given that it travels such that it just clears two walls of height h and distances 2h and 4h from O respectively. (a) find the tangent of the angle θ (b) The time of flight of the ball (c) The range of the ball.

$$\:\mathrm{A}\:\mathrm{ball}\:\mathrm{is}\:\mathrm{projected}\:\mathrm{from}\:\mathrm{a}\:\mathrm{point}\:\mathrm{O}\:\mathrm{with}\:\mathrm{an}\:\mathrm{initial} \\ $$$$\mathrm{velocity}\:{u}\:\mathrm{and}\:\mathrm{angle}\:\theta\:\mathrm{with}\:\mathrm{the}\:\mathrm{horizontal}\:\mathrm{ground}. \\ $$$$\:\mathrm{Given}\:\mathrm{that}\:\mathrm{it}\:\mathrm{travels}\:\mathrm{such}\:\mathrm{that}\:\mathrm{it}\:\mathrm{just}\:\mathrm{clears}\:\mathrm{two}\:\mathrm{walls} \\ $$$$\mathrm{of}\:\mathrm{height}\:{h}\:\mathrm{and}\:\mathrm{distances}\:\mathrm{2}{h}\:\mathrm{and}\:\mathrm{4}{h}\:\mathrm{from}\:\mathrm{O}\:\mathrm{respectively}. \\ $$$$\left(\mathrm{a}\right)\:\mathrm{find}\:\mathrm{the}\:\mathrm{tangent}\:\mathrm{of}\:\mathrm{the}\:\mathrm{angle}\:\theta \\ $$$$\:\left(\mathrm{b}\right)\:\mathrm{The}\:\mathrm{time}\:\mathrm{of}\:\mathrm{flight}\:\mathrm{of}\:\mathrm{the}\:\mathrm{ball} \\ $$$$\left(\mathrm{c}\right)\:\mathrm{The}\:\mathrm{range}\:\mathrm{of}\:\mathrm{the}\:\mathrm{ball}. \\ $$

Question Number 89318 Answers: 2 Comments: 0

∫(dx/(sin^3 (2x)+cos^3 (2x)))

$$\int\frac{{dx}}{{sin}^{\mathrm{3}} \left(\mathrm{2}{x}\right)+{cos}^{\mathrm{3}} \left(\mathrm{2}{x}\right)} \\ $$

Question Number 89317 Answers: 1 Comments: 2

find ∫ (dx/((x+(√(x−1)))^2 ))

$${find}\:\int\:\:\:\:\:\frac{{dx}}{\left({x}+\sqrt{{x}−\mathrm{1}}\right)^{\mathrm{2}} } \\ $$

Question Number 89316 Answers: 0 Comments: 0

calculate Σ_(n=2) ^∞ ((2n+1)/((n−1)^3 (n+1)^3 ))

$${calculate}\:\sum_{{n}=\mathrm{2}} ^{\infty} \:\:\:\frac{\mathrm{2}{n}+\mathrm{1}}{\left({n}−\mathrm{1}\right)^{\mathrm{3}} \left({n}+\mathrm{1}\right)^{\mathrm{3}} } \\ $$

Question Number 89315 Answers: 0 Comments: 0

calculate ∫∫_([0,1]^2 ) ((arctan(xy))/((x+y)^2 ))dxdy

$${calculate}\:\int\int_{\left[\mathrm{0},\mathrm{1}\right]^{\mathrm{2}} } \:\:\:\frac{{arctan}\left({xy}\right)}{\left({x}+{y}\right)^{\mathrm{2}} }{dxdy} \\ $$

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