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Question Number 29581 Answers: 1 Comments: 0

Let x = 4sin^2 10^o +4sin^2 50^o cos20^o +cos80^o and y = cos^2 (π/5)+cos^2 ((2π)/(15))+cos^2 ((8π)/(15)). find x+y ?

$${Let}\:{x}\:=\:\mathrm{4}{sin}^{\mathrm{2}} \mathrm{10}^{{o}} +\mathrm{4}{sin}^{\mathrm{2}} \mathrm{50}^{{o}} {cos}\mathrm{20}^{{o}} +{cos}\mathrm{80}^{{o}} \\ $$$${and}\:{y}\:=\:{cos}^{\mathrm{2}} \:\frac{\pi}{\mathrm{5}}+{cos}^{\mathrm{2}} \frac{\mathrm{2}\pi}{\mathrm{15}}+{cos}^{\mathrm{2}} \frac{\mathrm{8}\pi}{\mathrm{15}}. \\ $$$${find}\:{x}+{y}\:? \\ $$

Question Number 29687 Answers: 1 Comments: 4

Question Number 29592 Answers: 0 Comments: 0

y′=xy^2 +2y+1

$${y}'={xy}^{\mathrm{2}} +\mathrm{2}{y}+\mathrm{1} \\ $$

Question Number 29574 Answers: 0 Comments: 2

∫x^6 −1/x^2 −1dx

$$\int{x}^{\mathrm{6}} −\mathrm{1}/{x}^{\mathrm{2}} −\mathrm{1}{dx} \\ $$

Question Number 29573 Answers: 0 Comments: 2

derive the equation of a chain of length l mass m hanging between two points x distance apart.

$${derive}\:{the}\:{equation}\:{of}\:{a}\:{chain} \\ $$$${of}\:{length}\:{l}\:{mass}\:{m}\:{hanging} \\ $$$${between}\:{two}\:{points}\:{x}\:{distance} \\ $$$${apart}. \\ $$

Question Number 29559 Answers: 1 Comments: 6

Question Number 29554 Answers: 0 Comments: 1

let give f(x)= x^2 cos((1/x^2 )) if x∈]0,1] but its derivative f^′ is not integrable on ]0,1].

$$\left.{l}\left.{et}\:{give}\:{f}\left({x}\right)=\:{x}^{\mathrm{2}} {cos}\left(\frac{\mathrm{1}}{{x}^{\mathrm{2}} }\right)\:{if}\:{x}\in\right]\mathrm{0},\mathrm{1}\right]\:{but}\:{its}\:{derivative}\:{f}^{'} \\ $$$$\left.{i}\left.{s}\:{not}\:{integrable}\:{on}\:\right]\mathrm{0},\mathrm{1}\right]. \\ $$

Question Number 29553 Answers: 0 Comments: 0

let put I(x)= ∫_x ^(+∞) ((sin^3 t)/t^2 )dt with x>0 find lim_(x→0^+ ) I(x) 2) find ∫_0 ^∞ ((sin^3 t)/t^2 )dt .

$${let}\:{put}\:\:{I}\left({x}\right)=\:\int_{{x}} ^{+\infty} \:\frac{{sin}^{\mathrm{3}} {t}}{{t}^{\mathrm{2}} }{dt}\:\:{with}\:{x}>\mathrm{0} \\ $$$${find}\:{lim}_{{x}\rightarrow\mathrm{0}^{+} } \:\:{I}\left({x}\right) \\ $$$$\left.\mathrm{2}\right)\:{find}\:\int_{\mathrm{0}} ^{\infty} \:\:\frac{{sin}^{\mathrm{3}} {t}}{{t}^{\mathrm{2}} }{dt}\:. \\ $$

Question Number 29552 Answers: 1 Comments: 1

find ∫_0 ^∞ (((√(x+1)) −1)/(x(x+1)))dx .

$${find}\:\int_{\mathrm{0}} ^{\infty} \:\:\frac{\sqrt{{x}+\mathrm{1}}\:−\mathrm{1}}{{x}\left({x}+\mathrm{1}\right)}{dx}\:. \\ $$

Question Number 29551 Answers: 0 Comments: 1

find the value of ∫_0 ^∞ ((arctan(2x)−arctanx)/x)dx.

$${find}\:{the}\:{value}\:{of}\:\int_{\mathrm{0}} ^{\infty} \:\:\:\:\frac{{arctan}\left(\mathrm{2}{x}\right)−{arctanx}}{{x}}{dx}. \\ $$

Question Number 29550 Answers: 1 Comments: 0

Question Number 29541 Answers: 0 Comments: 4

Question Number 29536 Answers: 1 Comments: 5

Find eccentricity of the ellipse 7x^2 +7y^2 +2xy+10x−10y−7=0 ?

$${Find}\:{eccentricity}\:{of}\:{the}\:{ellipse} \\ $$$$\mathrm{7}{x}^{\mathrm{2}} +\mathrm{7}{y}^{\mathrm{2}} +\mathrm{2}{xy}+\mathrm{10}{x}−\mathrm{10}{y}−\mathrm{7}=\mathrm{0}\:? \\ $$

Question Number 29522 Answers: 1 Comments: 7

Question Number 29520 Answers: 1 Comments: 0

to make an open fish tank a glass sheet of 2mm gauge is used .the outer length ,breadth and height are 60.4 , 40.4, 40.2 respectively .how much maximum volume of water will be contained in it ?

$$\mathrm{to}\:\mathrm{make}\:\mathrm{an}\:\mathrm{open}\:\mathrm{fish}\:\mathrm{tank}\:\mathrm{a}\:\mathrm{glass}\:\mathrm{sheet}\:\mathrm{of}\: \\ $$$$\mathrm{2mm}\:\mathrm{gauge}\:\mathrm{is}\:\mathrm{used}\:.\mathrm{the}\:\mathrm{outer}\:\mathrm{length} \\ $$$$,\mathrm{breadth}\:\mathrm{and}\:\mathrm{height}\:\mathrm{are}\:\mathrm{60}.\mathrm{4}\:,\:\mathrm{40}.\mathrm{4},\: \\ $$$$\mathrm{40}.\mathrm{2}\:\mathrm{respectively}\:.\mathrm{how}\:\mathrm{much}\:\mathrm{maximum} \\ $$$$\mathrm{volume}\:\mathrm{of}\:\mathrm{water}\:\mathrm{will}\:\mathrm{be}\:\mathrm{contained}\:\mathrm{in}\:\mathrm{it}\:? \\ $$$$ \\ $$

Question Number 29517 Answers: 0 Comments: 1

∫x^6 −1/x^2 +1

$$\int\mathrm{x}^{\mathrm{6}} −\mathrm{1}/\mathrm{x}^{\mathrm{2}} +\mathrm{1} \\ $$

Question Number 29511 Answers: 0 Comments: 1

find lim_(n→+∞) (((n!)^2 )/((2n)!)) .

$${find}\:{lim}_{{n}\rightarrow+\infty} \:\:\:\:\:\:\frac{\left({n}!\right)^{\mathrm{2}} }{\left(\mathrm{2}{n}\right)!}\:. \\ $$

Question Number 29510 Answers: 0 Comments: 1

let w_n = (1/n)( 1 +e^(1/n) +e^(2/n) + ... e^((n−1)/n) ) find lim_(n→+∞) w_n .

$${let}\:\:\:{w}_{{n}} =\:\frac{\mathrm{1}}{{n}}\left(\:\mathrm{1}\:+{e}^{\frac{\mathrm{1}}{{n}}} \:+{e}^{\frac{\mathrm{2}}{{n}}} \:+\:...\:{e}^{\frac{{n}−\mathrm{1}}{{n}}} \right)\:{find}\:{lim}_{{n}\rightarrow+\infty} {w}_{{n}} . \\ $$

Question Number 29509 Answers: 0 Comments: 1

find lim_(n→+ ∞) ((n!)/2^(n−1) ) .

$${find}\:{lim}_{{n}\rightarrow+\:\infty} \:\:\:\:\frac{{n}!}{\mathrm{2}^{{n}−\mathrm{1}} }\:. \\ $$

Question Number 29508 Answers: 0 Comments: 0

x and y are elements from R wich verify (x+(√(x^2 +1)))(y +(√(y^2 +1)))= 1 find x+y .

$${x}\:{and}\:{y}\:{are}\:{elements}\:{from}\:{R}\:{wich}\:{verify} \\ $$$$\left({x}+\sqrt{{x}^{\mathrm{2}} +\mathrm{1}}\right)\left({y}\:+\sqrt{{y}^{\mathrm{2}} +\mathrm{1}}\right)=\:\mathrm{1}\:\:{find}\:{x}+{y}\:. \\ $$

Question Number 29507 Answers: 0 Comments: 0

solve [(√x) ] −x+1=0

$${solve}\:\:\:\left[\sqrt{{x}}\:\right]\:−{x}+\mathrm{1}=\mathrm{0} \\ $$

Question Number 29506 Answers: 0 Comments: 1

le give A_n = ∫_0 ^(π/2) ((sin((2n−1)x))/(sinx))dx and B_n =∫_0 ^(π/2) ((sin^2 (nx))/(sin^2 x))dx 1)calculate A_n 2)prove that B_(n+1) −B_n = A_(n+1) .then find B_n .

$${le}\:{give}\:\:{A}_{{n}} =\:\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \:\:\frac{{sin}\left(\left(\mathrm{2}{n}−\mathrm{1}\right){x}\right)}{{sinx}}{dx}\:{and}\:{B}_{{n}} =\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \:\frac{{sin}^{\mathrm{2}} \left({nx}\right)}{{sin}^{\mathrm{2}} {x}}{dx} \\ $$$$\left.\mathrm{1}\right){calculate}\:{A}_{{n}} \\ $$$$\left.\mathrm{2}\right){prove}\:{that}\:{B}_{{n}+\mathrm{1}} −{B}_{{n}} =\:{A}_{{n}+\mathrm{1}} .{then}\:{find}\:{B}_{{n}} . \\ $$

Question Number 29505 Answers: 0 Comments: 3

let give u_n = Σ_(k=0) ^∞ (1/((k+1)^2 2^k )) find lim_(n→∞) u_n .

$${let}\:{give}\:{u}_{{n}} =\:\:\sum_{{k}=\mathrm{0}} ^{\infty} \:\:\:\frac{\mathrm{1}}{\left({k}+\mathrm{1}\right)^{\mathrm{2}} \:\mathrm{2}^{{k}} }\:\:{find}\:\:{lim}_{{n}\rightarrow\infty} {u}_{{n}} \:\:. \\ $$

Question Number 29503 Answers: 0 Comments: 0

let U_(n,p) = Σ_(k=1) ^n (1/((n+k)^(p+1) )) with n,p from N^★ 1) calculate lim_(n→+∞) U_(n,p) for p≥2 2)prove that U_(n,1) is convergent 3) let V_n = Σ_(k=1) ^n sin((1/((n+k)^2 ))) find lim_∞ V_n .

$${let}\:{U}_{{n},{p}} =\:\sum_{{k}=\mathrm{1}} ^{{n}} \:\:\frac{\mathrm{1}}{\left({n}+{k}\right)^{{p}+\mathrm{1}} }\:{with}\:{n},{p}\:{from}\:{N}^{\bigstar} \\ $$$$\left.\mathrm{1}\right)\:{calculate}\:{lim}_{{n}\rightarrow+\infty} {U}_{{n},{p}} \:{for}\:{p}\geqslant\mathrm{2} \\ $$$$\left.\mathrm{2}\right){prove}\:{that}\:{U}_{{n},\mathrm{1}} \:{is}\:{convergent} \\ $$$$\left.\mathrm{3}\right)\:{let}\:{V}_{{n}} =\:\sum_{{k}=\mathrm{1}} ^{{n}} \:{sin}\left(\frac{\mathrm{1}}{\left({n}+{k}\right)^{\mathrm{2}} }\right)\:{find}\:{lim}_{\infty} \:{V}_{{n}} \:. \\ $$

Question Number 29501 Answers: 0 Comments: 0

find Σ_(n=1) ^∞ (1/(n^2 (n+1)^2 (n+2)^2 )).

$${find}\:\:\sum_{{n}=\mathrm{1}} ^{\infty} \:\:\:\frac{\mathrm{1}}{{n}^{\mathrm{2}} \left({n}+\mathrm{1}\right)^{\mathrm{2}} \left({n}+\mathrm{2}\right)^{\mathrm{2}} }. \\ $$

Question Number 29500 Answers: 0 Comments: 1

nature of the sequence u_n = Σ_(k=2) ^n (((−1)^k )/(kln(k))) .

$${nature}\:{of}\:{the}\:{sequence}\:\:{u}_{{n}} =\:\sum_{{k}=\mathrm{2}} ^{{n}} \:\frac{\left(−\mathrm{1}\right)^{{k}} }{{kln}\left({k}\right)}\:. \\ $$

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