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Question Number 48509 Answers: 0 Comments: 0

For every natural numbers n Find the value of Σ_(0≤j≤i≤n) (((−1)^j )/((n − i)! j!))

$${For}\:\:{every}\:\:{natural}\:\:{numbers}\:\:{n}\:\: \\ $$$${Find}\:\:\:{the}\:\:{value}\:\:{of} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\underset{\mathrm{0}\leqslant{j}\leqslant{i}\leqslant{n}} {\sum}\:\:\frac{\left(−\mathrm{1}\right)^{{j}} }{\left({n}\:−\:{i}\right)!\:{j}!} \\ $$

Question Number 48506 Answers: 0 Comments: 0

let S_n =Σ_(k=0) ^∞ (((−1)^k )/(2k+1)) 1)prove that (π/4) −S_n =(−1)^(n+1) ∫_0 ^1 (t^(2n+2) /(1+t^2 ))dt 2) conclude lim_(n→+∞) S_n .

$${let}\:\:{S}_{{n}} =\sum_{{k}=\mathrm{0}} ^{\infty} \:\frac{\left(−\mathrm{1}\right)^{{k}} }{\mathrm{2}{k}+\mathrm{1}} \\ $$$$\left.\mathrm{1}\right){prove}\:{that}\:\frac{\pi}{\mathrm{4}}\:−{S}_{{n}} =\left(−\mathrm{1}\right)^{{n}+\mathrm{1}} \:\int_{\mathrm{0}} ^{\mathrm{1}} \:\frac{{t}^{\mathrm{2}{n}+\mathrm{2}} }{\mathrm{1}+{t}^{\mathrm{2}} }{dt} \\ $$$$\left.\mathrm{2}\right)\:{conclude}\:{lim}_{{n}\rightarrow+\infty} \:\:{S}_{{n}} . \\ $$

Question Number 48501 Answers: 1 Comments: 0

Question Number 48500 Answers: 0 Comments: 2

Question Number 48498 Answers: 2 Comments: 4

find A_n = ∫_0 ^(π/4) cos^n xdx and B_n =∫_0 ^(π/4) sin^n xdx 2) find ∫_0 ^(π/4) cos^6 xdx and ∫_0 ^(π/4) sin^6 xdx .

$${find}\:{A}_{{n}} =\:\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{4}}} \:{cos}^{{n}} {xdx}\:\:{and}\:{B}_{{n}} =\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{4}}} \:{sin}^{{n}} {xdx} \\ $$$$\left.\mathrm{2}\right)\:{find}\:\:\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{4}}} \:{cos}^{\mathrm{6}} {xdx}\:{and}\:\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{4}}} \:{sin}^{\mathrm{6}} {xdx}\:. \\ $$

Question Number 48497 Answers: 0 Comments: 5

let f(x)=∫_0 ^1 ((ln(1+xt^2 ))/(1+t^2 ))dt 1) find a xplicit form of f(x) 2) developp f at integr serie 3)find the value of ∫_0 ^1 ((ln(1+t^2 ))/(1+t^2 ))dt 4)find the value of ∫_0 ^1 ((ln(1+2t^2 ))/(1+t^2 ))dt

$${let}\:{f}\left({x}\right)=\int_{\mathrm{0}} ^{\mathrm{1}} \:\:\frac{{ln}\left(\mathrm{1}+{xt}^{\mathrm{2}} \right)}{\mathrm{1}+{t}^{\mathrm{2}} }{dt} \\ $$$$\left.\mathrm{1}\right)\:{find}\:{a}\:{xplicit}\:{form}\:{of}\:{f}\left({x}\right) \\ $$$$\left.\mathrm{2}\right)\:{developp}\:{f}\:{at}\:{integr}\:{serie}\: \\ $$$$\left.\mathrm{3}\right){find}\:{the}\:{value}\:{of}\:\int_{\mathrm{0}} ^{\mathrm{1}} \:\frac{{ln}\left(\mathrm{1}+{t}^{\mathrm{2}} \right)}{\mathrm{1}+{t}^{\mathrm{2}} }{dt} \\ $$$$\left.\mathrm{4}\right){find}\:{the}\:{value}\:{of}\:\int_{\mathrm{0}} ^{\mathrm{1}} \:\frac{{ln}\left(\mathrm{1}+\mathrm{2}{t}^{\mathrm{2}} \right)}{\mathrm{1}+{t}^{\mathrm{2}} }{dt} \\ $$$$ \\ $$

Question Number 48496 Answers: 0 Comments: 0

find f(x) =∫ ((ln(1+xt^2 ))/(1+t^2 ))dt

$${find}\:{f}\left({x}\right)\:=\int\:\:\frac{{ln}\left(\mathrm{1}+{xt}^{\mathrm{2}} \right)}{\mathrm{1}+{t}^{\mathrm{2}} }{dt} \\ $$

Question Number 48493 Answers: 2 Comments: 1

let S_n =Σ_(k=1) ^n (k^2 /((2k−1)(2k+1))) 1) determine S_n interms of n 2) find lim_(n→+∞) S_n

$${let}\:{S}_{{n}} =\sum_{{k}=\mathrm{1}} ^{{n}} \:\:\frac{{k}^{\mathrm{2}} }{\left(\mathrm{2}{k}−\mathrm{1}\right)\left(\mathrm{2}{k}+\mathrm{1}\right)} \\ $$$$\left.\mathrm{1}\right)\:{determine}\:{S}_{{n}} \:{interms}\:{of}\:{n} \\ $$$$\left.\mathrm{2}\right)\:{find}\:{lim}_{{n}\rightarrow+\infty} \:{S}_{{n}} \\ $$

Question Number 48495 Answers: 1 Comments: 1

1)calculate I =∫ ((ln(1+t))/(1+t))dt 2) find ∫_0 ^1 ((ln(1+t))/(1+t))dt

$$\left.\mathrm{1}\right){calculate}\:\:{I}\:=\int\:\frac{{ln}\left(\mathrm{1}+{t}\right)}{\mathrm{1}+{t}}{dt} \\ $$$$\left.\mathrm{2}\right)\:{find}\:\:\:\int_{\mathrm{0}} ^{\mathrm{1}} \:\:\frac{{ln}\left(\mathrm{1}+{t}\right)}{\mathrm{1}+{t}}{dt} \\ $$

Question Number 48494 Answers: 1 Comments: 0

find A_n =∫_0 ^(π/2) ((1−cos(n+1)x)/(2sin((x/2))))dx .

$${find}\:{A}_{{n}} =\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \:\frac{\mathrm{1}−{cos}\left({n}+\mathrm{1}\right){x}}{\mathrm{2}{sin}\left(\frac{{x}}{\mathrm{2}}\right)}{dx}\:. \\ $$

Question Number 48491 Answers: 0 Comments: 0

prove that ∫_0 ^∞ (((1+t)^(−(3/4)) −(1+t)^(−(1/4)) )/t)dt is convergent and find its value .

$${prove}\:{that}\:\int_{\mathrm{0}} ^{\infty} \:\:\frac{\left(\mathrm{1}+{t}\right)^{−\frac{\mathrm{3}}{\mathrm{4}}} −\left(\mathrm{1}+{t}\right)^{−\frac{\mathrm{1}}{\mathrm{4}}} }{{t}}{dt}\:{is}\:{convergent}\:{and}\:{find}\:{its}\:{value}\:. \\ $$

Question Number 48489 Answers: 0 Comments: 0

Question Number 48484 Answers: 1 Comments: 0

Question Number 48482 Answers: 0 Comments: 0

(at−h)^2 +((a/t)−k)^2 =R^( 2) where a, h, k, R are constants. Then find s^2 =(t_1 −t_2 )^2 (1+(1/(t_1 ^2 t_2 ^2 ))) where t_1 , t_2 are roots of eq. at top.

$$\left({at}−{h}\right)^{\mathrm{2}} +\left(\frac{{a}}{{t}}−{k}\right)^{\mathrm{2}} ={R}^{\:\mathrm{2}} \\ $$$${where}\:\:\:{a},\:{h},\:{k},\:{R}\:{are}\:{constants}. \\ $$$${Then}\:{find}\: \\ $$$$\:\:\:{s}^{\mathrm{2}} \:=\left({t}_{\mathrm{1}} −{t}_{\mathrm{2}} \right)^{\mathrm{2}} \left(\mathrm{1}+\frac{\mathrm{1}}{{t}_{\mathrm{1}} ^{\mathrm{2}} {t}_{\mathrm{2}} ^{\mathrm{2}} }\right)\: \\ $$$${where}\:{t}_{\mathrm{1}} ,\:{t}_{\mathrm{2}} \:{are}\:{roots}\:{of}\:{eq}.\:{at}\:{top}. \\ $$

Question Number 48474 Answers: 2 Comments: 0

evaluate ∫_0 ^π sin^2 xdx

$$\boldsymbol{\mathrm{evaluate}}\:\int_{\mathrm{0}} ^{\pi} \boldsymbol{\mathrm{sin}}^{\mathrm{2}} \boldsymbol{{x}\mathrm{d}{x}} \\ $$

Question Number 48466 Answers: 1 Comments: 0

Question Number 48460 Answers: 1 Comments: 2

A point source has a distance d to the center of a big sphere with radius R. An other smaller sphere with radius r is placed between the point source and the big sphere. If the distance between the two spheres is constant, say it′s c. Find the maximal shadow area of the small sphere on the surface of the big sphere. Find also the minimal complete shadow of the small sphere on the surface of the big sphere. Assume the small sphere is much smaller than the big sphere such that the big sphere will never completely stay in the shadow of the small sphere.

$${A}\:{point}\:{source}\:{has}\:{a}\:{distance}\:{d} \\ $$$${to}\:{the}\:{center}\:{of}\:{a}\:{big}\:{sphere}\:{with}\:{radius} \\ $$$${R}.\:\:{An}\:{other}\:{smaller}\:{sphere}\:{with}\:{radius} \\ $$$${r}\:{is}\:{placed}\:{between}\:{the}\:{point}\:{source} \\ $$$${and}\:{the}\:{big}\:{sphere}.\:{If}\:{the}\:{distance} \\ $$$${between}\:{the}\:{two}\:{spheres}\:{is}\:{constant}, \\ $$$${say}\:{it}'{s}\:{c}.\: \\ $$$${Find}\:{the}\:{maximal}\:{shadow}\:{area}\:{of}\:{the} \\ $$$${small}\:{sphere}\:{on}\:{the}\:{surface}\:{of}\:{the} \\ $$$${big}\:{sphere}.\:{Find}\:{also}\:{the}\:{minimal} \\ $$$${complete}\:{shadow}\:{of}\:{the}\:{small}\:{sphere} \\ $$$${on}\:{the}\:{surface}\:{of}\:{the}\:{big}\:{sphere}. \\ $$$$ \\ $$$${Assume}\:{the}\:{small}\:{sphere}\:{is}\:{much} \\ $$$${smaller}\:{than}\:{the}\:{big}\:{sphere}\:{such}\:{that} \\ $$$${the}\:{big}\:{sphere}\:{will}\:{never}\:{completely}\:{stay} \\ $$$${in}\:{the}\:{shadow}\:{of}\:{the}\:{small}\:{sphere}. \\ $$

Question Number 48458 Answers: 1 Comments: 1