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Question Number 32340 Answers: 0 Comments: 0

find the value of ∫_0 ^∞ ((sin^3 t)/t^2 ) dt .

$${find}\:{the}\:{value}\:{of}\:\:\int_{\mathrm{0}} ^{\infty} \:\:\:\frac{{sin}^{\mathrm{3}} {t}}{{t}^{\mathrm{2}} }\:{dt}\:. \\ $$

Question Number 32339 Answers: 0 Comments: 1

calculate ∫_0 ^(+∞) ((th(3x) −th(2x))/x) dx .

$${calculate}\:\:\int_{\mathrm{0}} ^{+\infty} \:\:\:\frac{{th}\left(\mathrm{3}{x}\right)\:−{th}\left(\mathrm{2}{x}\right)}{{x}}\:{dx}\:. \\ $$

Question Number 32338 Answers: 0 Comments: 0

find the value of ∫_0 ^1 ((ln(t))/((1+t)(√(1−t^2 )))) dt.

$${find}\:{the}\:{value}\:{of}\:\:\int_{\mathrm{0}} ^{\mathrm{1}} \:\:\:\:\frac{{ln}\left({t}\right)}{\left(\mathrm{1}+{t}\right)\sqrt{\mathrm{1}−{t}^{\mathrm{2}} }}\:{dt}. \\ $$

Question Number 32337 Answers: 0 Comments: 0

1)calculate ∫_a ^(+∞) (dx/((1+x^2 )(√(x^2 −a^2 )))) with a>0 2) find the value of ∫_2 ^(+∞) (dx/((1+x^2 )(√(x^2 −4)))) .

$$\left.\mathrm{1}\right){calculate}\:\int_{{a}} ^{+\infty} \:\:\:\:\:\frac{{dx}}{\left(\mathrm{1}+{x}^{\mathrm{2}} \right)\sqrt{{x}^{\mathrm{2}} \:−{a}^{\mathrm{2}} }}\:\:{with}\:{a}>\mathrm{0} \\ $$$$\left.\mathrm{2}\right)\:{find}\:{the}\:{value}\:{of}\:\:\:\int_{\mathrm{2}} ^{+\infty} \:\:\:\:\:\:\frac{{dx}}{\left(\mathrm{1}+{x}^{\mathrm{2}} \right)\sqrt{{x}^{\mathrm{2}} \:−\mathrm{4}}}\:\:. \\ $$

Question Number 32335 Answers: 0 Comments: 0

let t≥0 and f(t) =(t/(√(1+t))) .prove that the sequence S_n = Σ_(k=1) ^n f((k/n^2 )) converges and find its limit.

$${let}\:{t}\geqslant\mathrm{0}\:{and}\:{f}\left({t}\right)\:=\frac{{t}}{\sqrt{\mathrm{1}+{t}}}\:.{prove}\:{that}\:{the}\:{sequence} \\ $$$${S}_{{n}} =\:\sum_{{k}=\mathrm{1}} ^{{n}} \:{f}\left(\frac{{k}}{{n}^{\mathrm{2}} }\right)\:\:{converges}\:{and}\:{find}\:{its}\:{limit}. \\ $$

Question Number 32334 Answers: 0 Comments: 1

p is apolynomial having n roots (x_i ) with x_i ≠xj for i≠j calculate Σ_(k=1) ^n (1/(1−x_k )) .

$${p}\:{is}\:{apolynomial}\:{having}\:{n}\:{roots}\:\:\left({x}_{{i}} \right)\:{with}\:{x}_{{i}} \neq{xj}\:{for} \\ $$$${i}\neq{j}\:\:{calculate}\:\:\sum_{{k}=\mathrm{1}} ^{{n}} \:\:\:\frac{\mathrm{1}}{\mathrm{1}−{x}_{{k}} }\:. \\ $$

Question Number 32333 Answers: 0 Comments: 1

decompose F(x) = (1/((1−x)^2 (1−x^2 ))) inside R(x).

$${decompose}\:{F}\left({x}\right)\:=\:\frac{\mathrm{1}}{\left(\mathrm{1}−{x}\right)^{\mathrm{2}} \left(\mathrm{1}−{x}^{\mathrm{2}} \right)}\:{inside}\:{R}\left({x}\right). \\ $$

Question Number 32332 Answers: 0 Comments: 0

calculate Σ_(p=1) ^n (p/(1+p^2 +p^4 )) .

$${calculate}\:\:\sum_{{p}=\mathrm{1}} ^{{n}} \:\:\:\frac{{p}}{\mathrm{1}+{p}^{\mathrm{2}} \:+{p}^{\mathrm{4}} }\:\:. \\ $$

Question Number 32331 Answers: 0 Comments: 0

p is a polynomial having n simples roots (x_i )_(1≤i≤n) prove that Σ_(k=1) ^n (1/(p^′ (x_k ))) =0

$${p}\:{is}\:{a}\:{polynomial}\:{having}\:{n}\:{simples}\:{roots}\:\left({x}_{{i}} \right)_{\mathrm{1}\leqslant{i}\leqslant{n}} \\ $$$${prove}\:{that}\:\:\sum_{{k}=\mathrm{1}} ^{{n}} \:\:\frac{\mathrm{1}}{{p}^{'} \left({x}_{{k}} \right)}\:=\mathrm{0} \\ $$

Question Number 32330 Answers: 0 Comments: 1

let p_n (x)=(x+1)^(6n+1) −x^(6n+1) −1 with n integr prove that ∀n (x^2 +x+1)^2 divide p_n (x).

$${let}\:{p}_{{n}} \left({x}\right)=\left({x}+\mathrm{1}\right)^{\mathrm{6}{n}+\mathrm{1}} \:−{x}^{\mathrm{6}{n}+\mathrm{1}} \:−\mathrm{1}\:{with}\:{n}\:{integr} \\ $$$${prove}\:{that}\:\forall{n}\:\:\left({x}^{\mathrm{2}} +{x}+\mathrm{1}\right)^{\mathrm{2}} \:{divide}\:{p}_{{n}} \left({x}\right). \\ $$

Question Number 32329 Answers: 2 Comments: 0

The smallest number which must be subtracted from 3400 to make it a perfect cube is _____.

$$\mathrm{The}\:\mathrm{smallest}\:\mathrm{number}\:\mathrm{which}\:\mathrm{must}\:\mathrm{be} \\ $$$$\mathrm{subtracted}\:\mathrm{from}\:\mathrm{3400}\:\mathrm{to}\:\mathrm{make}\:\mathrm{it}\:\mathrm{a}\: \\ $$$$\mathrm{perfect}\:\mathrm{cube}\:\mathrm{is}\:\_\_\_\_\_. \\ $$

Question Number 32328 Answers: 0 Comments: 0

prove that 2^(n+1) divide A_n =[(1+(√3) )^(2n+1) ].

$${prove}\:{that}\:\mathrm{2}^{{n}+\mathrm{1}} \:{divide}\:\:{A}_{{n}} =\left[\left(\mathrm{1}+\sqrt{\mathrm{3}}\:\right)^{\mathrm{2}{n}+\mathrm{1}} \right]. \\ $$

Question Number 32326 Answers: 0 Comments: 0

simplify Σ_(k=p) ^(2p) (C_k ^p /2^k ) .

$${simplify}\:\:\sum_{{k}={p}} ^{\mathrm{2}{p}} \:\:\:\:\:\frac{{C}_{{k}} ^{{p}} }{\mathrm{2}^{{k}} }\:. \\ $$

Question Number 32323 Answers: 1 Comments: 0

Given f(x) = (3/(16))(∫_0 ^1 f(x)dx)x^2 − (9/(10))(∫_0 ^2 f(x)dx)x + 2(∫_0 ^3 f(x)dx) + 4 Find f(x)

$$\mathrm{Given} \\ $$$${f}\left({x}\right)\:=\:\frac{\mathrm{3}}{\mathrm{16}}\left(\int_{\mathrm{0}} ^{\mathrm{1}} {f}\left({x}\right){dx}\right){x}^{\mathrm{2}} \:−\:\frac{\mathrm{9}}{\mathrm{10}}\left(\int_{\mathrm{0}} ^{\mathrm{2}} {f}\left({x}\right){dx}\right){x}\:+\:\mathrm{2}\left(\int_{\mathrm{0}} ^{\mathrm{3}} {f}\left({x}\right){dx}\right)\:+\:\mathrm{4} \\ $$$$\mathrm{Find}\:{f}\left({x}\right) \\ $$

Question Number 32316 Answers: 0 Comments: 4

A gas expands according to the law pv=constant,where p is the pressure and v is the volume of the gas.Initially,v=1000m^3 and p=40N/m^2 .If the pressure is decreased at the rate of 5Nm^(−2) min^(−1) find the rate at which the gas is expanding when its volume is 2000m^(3.)

$${A}\:{gas}\:{expands}\:{according}\:{to}\:{the} \\ $$$${law}\:{pv}={constant},{where}\:{p}\:{is}\:{the} \\ $$$${pressure}\:{and}\:{v}\:{is}\:{the}\:{volume}\:{of} \\ $$$${the}\:{gas}.{Initially},{v}=\mathrm{1000}{m}^{\mathrm{3}} \:{and} \\ $$$${p}=\mathrm{40}{N}/{m}^{\mathrm{2}} .{If}\:{the}\:{pressure}\:{is} \\ $$$${decreased}\:{at}\:{the}\:{rate}\:{of}\:\mathrm{5}{Nm}^{−\mathrm{2}} {min}^{−\mathrm{1}} \\ $$$${find}\:{the}\:{rate}\:{at}\:{which}\:{the}\:{gas}\:{is} \\ $$$${expanding}\:{when}\:{its}\:{volume}\:{is} \\ $$$$\mathrm{2000}{m}^{\mathrm{3}.} \\ $$

Question Number 32313 Answers: 0 Comments: 0

If y=sinx then show that (d^n y/dx^n )=sin(x+((nπ)/2))

$${If}\:{y}={sinx}\:{then}\:{show}\:{that} \\ $$$$\frac{{d}^{{n}} {y}}{{dx}^{{n}} }={sin}\left({x}+\frac{{n}\pi}{\mathrm{2}}\right) \\ $$

Question Number 32312 Answers: 1 Comments: 1

A small ball is dropped from a height of 1m into a horizontal floor.Each time it rebounces to 3/5 of the height it has fallen. a)show that when the ball strikes the ground for the third time ,it has travelled a distance of 2.92m b)Show that the total distance travelled by the ball cant exceed 4m.

$${A}\:{small}\:{ball}\:{is}\:{dropped}\:{from}\:{a}\: \\ $$$${height}\:{of}\:\mathrm{1}{m}\:{into}\:{a}\:{horizontal} \\ $$$${floor}.{Each}\:{time}\:{it}\:{rebounces}\:{to} \\ $$$$\mathrm{3}/\mathrm{5}\:{of}\:{the}\:{height}\:{it}\:{has}\:{fallen}. \\ $$$$\left.{a}\right){show}\:{that}\:{when}\:{the}\:{ball}\:{strikes} \\ $$$${the}\:{ground}\:{for}\:{the}\:{third}\:{time}\:,{it} \\ $$$${has}\:{travelled}\:{a}\:{distance}\:{of}\:\mathrm{2}.\mathrm{92}{m} \\ $$$$\left.{b}\right){Show}\:{that}\:{the}\:{total}\:{distance} \\ $$$${travelled}\:{by}\:{the}\:{ball}\:{cant}\:{exceed} \\ $$$$\mathrm{4}{m}. \\ $$

Question Number 32309 Answers: 0 Comments: 0