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Question Number 28280 Answers: 1 Comments: 4

Find dy/dx x^(2/3) (6−x)^(1/(3 )) to it simplest form

$${Find}\:{dy}/{dx} \\ $$$${x}^{\frac{\mathrm{2}}{\mathrm{3}}} \left(\mathrm{6}−{x}\right)^{\frac{\mathrm{1}}{\mathrm{3}\:}} \:{to}\:{it}\:{simplest}\:{form} \\ $$

Question Number 28278 Answers: 2 Comments: 1

Question Number 28275 Answers: 0 Comments: 2

Find area of the region [y]=[x] for x∈[2, 5] . [x] is greatest integer less than or equal to x .

$${Find}\:{area}\:{of}\:{the}\:{region} \\ $$$$\left[{y}\right]=\left[{x}\right]\:\:{for}\:\:{x}\in\left[\mathrm{2},\:\mathrm{5}\right]\:. \\ $$$$\left[{x}\right]\:{is}\:{greatest}\:{integer}\:{less}\:{than}\:{or} \\ $$$${equal}\:{to}\:{x}\:. \\ $$

Question Number 28268 Answers: 0 Comments: 1

find in terms of n the value of A_n = ∫_0 ^1 (1+x^2 )^(n/2) sin(narctanx)dx . ( n∈ N).

$$\:{find}\:{in}\:{terms}\:{of}\:{n}\:{the}\:{value}\:{of} \\ $$$${A}_{{n}} =\:\:\int_{\mathrm{0}} ^{\mathrm{1}} \left(\mathrm{1}+{x}^{\mathrm{2}} \right)^{\frac{{n}}{\mathrm{2}}} \:{sin}\left({narctanx}\right){dx}\:.\:\:\left(\:{n}\in\:{N}\right). \\ $$

Question Number 28267 Answers: 1 Comments: 1

let give the polynomial P(x)= (1/(2i))( (1+ix)^n −(1−ix)^n ) .find the roots of P(x) and factorize P(x).

$${let}\:{give}\:{the}\:{polynomial} \\ $$$${P}\left({x}\right)=\:\frac{\mathrm{1}}{\mathrm{2}{i}}\left(\:\left(\mathrm{1}+{ix}\right)^{{n}} \:−\left(\mathrm{1}−{ix}\right)^{{n}} \right)\:.{find}\:{the}\:{roots}\:{of}\:{P}\left({x}\right) \\ $$$${and}\:{factorize}\:{P}\left({x}\right). \\ $$

Question Number 28265 Answers: 0 Comments: 0

1) find P∈R[x] / P(sinx) =sin(2n+1)x 2) find the roots of P and degP 3) decompose (1/P) and prove that ((2n+1)/(sin(2n+1)x)) = Σ_(k=0) ^(2n) (((−1)^k cos(((kπ)/(2n+1))))/(sinx−sin (((kπ)/(2n+1)))))) .

$$\left.\mathrm{1}\right)\:\:{find}\:{P}\in{R}\left[{x}\right]\:/\:{P}\left({sinx}\right)\:={sin}\left(\mathrm{2}{n}+\mathrm{1}\right){x} \\ $$$$\left.\mathrm{2}\right)\:{find}\:{the}\:{roots}\:{of}\:{P}\:{and}\:{degP} \\ $$$$\left.\mathrm{3}\right)\:{decompose}\:\:\frac{\mathrm{1}}{{P}}\:\:{and}\:{prove}\:{that} \\ $$$$\frac{\mathrm{2}{n}+\mathrm{1}}{{sin}\left(\mathrm{2}{n}+\mathrm{1}\right){x}}\:=\:\sum_{{k}=\mathrm{0}} ^{\mathrm{2}{n}} \:\:\:\:\frac{\left(−\mathrm{1}\right)^{{k}} \:{cos}\left(\frac{{k}\pi}{\mathrm{2}{n}+\mathrm{1}}\right)}{{sinx}−{sin}\:\left(\frac{{k}\pi}{\left.\mathrm{2}{n}+\mathrm{1}\right)}\right)}\:\:. \\ $$

Question Number 28264 Answers: 0 Comments: 0

give the decomposition of F(x) = ((1 )/(Π_(k=1) ^n (x−k^2 ))) .

$${give}\:{the}\:{decomposition}\:{of}\: \\ $$$${F}\left({x}\right)\:\:\:=\:\:\:\:\:\:\frac{\mathrm{1}\:}{\prod_{{k}=\mathrm{1}} ^{{n}} \:\left({x}−{k}^{\mathrm{2}} \right)}\:. \\ $$

Question Number 28263 Answers: 0 Comments: 0

decompose F(x)= (1/((x−1)^2 (x^3 −1))) then calculate ∫_2 ^(+∝) F(x)dx.

$${decompose}\:{F}\left({x}\right)=\:\:\frac{\mathrm{1}}{\left({x}−\mathrm{1}\right)^{\mathrm{2}} \left({x}^{\mathrm{3}} −\mathrm{1}\right)}\:\:{then}\:{calculate} \\ $$$$\int_{\mathrm{2}} ^{+\propto} \:{F}\left({x}\right){dx}. \\ $$

Question Number 28262 Answers: 0 Comments: 1

find ∫_0 ^∞ (dx/(1+x^5 )) .

$${find}\:\int_{\mathrm{0}} ^{\infty} \:\:\frac{{dx}}{\mathrm{1}+{x}^{\mathrm{5}} }\:. \\ $$

Question Number 28261 Answers: 0 Comments: 0

let give the matrice ( 0 cosθ cos(2θ)) A= ( cosθ 0 cos(2θ) ) ( cos(θ) cos(2θ 0 ) and D_θ =det A solve inside R D_θ =0

$${let}\:{give}\:{the}\:{matrice} \\ $$$$\:\:\:\:\:\:\:\:\:\:\left(\:\:\:\mathrm{0}\:\:\:\:\:\:\:{cos}\theta\:\:\:\:\:\:{cos}\left(\mathrm{2}\theta\right)\right) \\ $$$${A}=\:\:\:\left(\:{cos}\theta\:\:\:\:\:\mathrm{0}\:\:\:\:\:\:\:\:\:{cos}\left(\mathrm{2}\theta\right)\:\right) \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\left(\:{cos}\left(\theta\right)\:{cos}\left(\mathrm{2}\theta\:\:\:\:\:\:\mathrm{0}\:\:\:\:\:\:\right)\right. \\ $$$${and}\:{D}_{\theta} \:\:={det}\:{A}\:\:{solve}\:{inside}\:{R}\:\:{D}_{\theta} =\mathrm{0} \\ $$$$\:\:\:\: \\ $$$$\: \\ $$

Question Number 28260 Answers: 0 Comments: 0

let give ( 1 1 −1) A= ( 1 1 1 ) ( −1 1 1 ) and the matrices I= ( 1 0 0 ) ( 0 1 1 ) ( 0 0 1 ) and J= ( 0 1 −1) ( 1 0 1). ( −1 1 0) 1) find J^2 and J^(−1) . 2) let put J^n = x_n I +y_n J .prove that x_(n+2 ) +x_(n+1) −2x_n =0 3) calculate J^n and A^n .

Question Number 28259 Answers: 0 Comments: 0

let give A= ( cosθ −sinθ ) ( sinθ cosθ ) 1) calculate^t A. A .prove that A is inversible and find A^(−1) 2) find A^n for n∈ N 3) developp (A +A^(−1) )^n then prove that 2^n cos^n θ = Σ_(k=0) ^n C_n ^k (n−2k)θ and Σ_(k=0) ^n C_n ^n sin(n−2k)θ =0 .

$${let}\:{give}\:\:\:\:{A}=\:\:\:\left(\:\:{cos}\theta\:\:\:\:\:\:\:−{sin}\theta\:\:\right) \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\left(\:{sin}\theta\:\:\:\:\:\:\:\:\:{cos}\theta\:\:\:\:\right) \\ $$$$\left.\mathrm{1}\right)\:\:{calculate}\:^{{t}} {A}.\:{A}\:\:.{prove}\:{that}\:{A}\:{is}\:{inversible}\:{and}\:{find} \\ $$$${A}^{−\mathrm{1}} \\ $$$$\left.\mathrm{2}\right)\:\:{find}\:\:{A}^{{n}} \:\:\:{for}\:{n}\in\:{N} \\ $$$$\left.\mathrm{3}\right)\:{developp}\:\left({A}\:+{A}^{−\mathrm{1}} \right)^{{n}} \:\:{then}\:{prove}\:{that} \\ $$$$\mathrm{2}^{{n}} \:{cos}^{{n}} \theta\:\:=\:\:\sum_{{k}=\mathrm{0}} ^{{n}} \:{C}_{{n}} ^{{k}} \left({n}−\mathrm{2}{k}\right)\theta\:\:{and} \\ $$$$\sum_{{k}=\mathrm{0}} ^{{n}} \:{C}_{{n}} ^{{n}} \:{sin}\left({n}−\mathrm{2}{k}\right)\theta\:\:=\mathrm{0}\:\:. \\ $$

Question Number 28258 Answers: 0 Comments: 1

let give A = ( 1 1 ) ( 2 −1) find e^(A ) and e^(−tA) .

$${let}\:{give}\:{A}\:\:=\:\left(\:\mathrm{1}\:\:\:\:\:\:\:\mathrm{1}\:\right) \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\left(\:\:\mathrm{2}\:\:\:\:−\mathrm{1}\right) \\ $$$$\:\:{find}\:\:{e}^{{A}\:} \:\:\:\:{and}\:\:\:{e}^{−{tA}} .\:\: \\ $$

Question Number 28257 Answers: 0 Comments: 0

let give ( 2 3 −3) A = ( −1 0 1) ( −1 1 0 ) find a diagoal matrice D and a inversible matrice P wich verify A = P.D.P^(−1) and calculate A^n .

Question Number 28248 Answers: 1 Comments: 0

Question Number 28247 Answers: 0 Comments: 1

find the value of ∫_1 ^(+∞) ((lnx)/(1+x^2 ))dx .

$${find}\:{the}\:{value}\:{of}\:\int_{\mathrm{1}} ^{+\infty} \:\:\frac{{lnx}}{\mathrm{1}+{x}^{\mathrm{2}} }{dx}\:. \\ $$

Question Number 28242 Answers: 0 Comments: 1

find the value of ∫_0 ^∞ e^(−x) lnxdx .

$${find}\:{the}\:{value}\:{of}\:\int_{\mathrm{0}} ^{\infty} {e}^{−{x}} {lnxdx}\:\:. \\ $$

Question Number 28241 Answers: 1 Comments: 1

Question Number 28240 Answers: 1 Comments: 0

Question Number 28255 Answers: 0 Comments: 0

let give p=(_(1 −2) ^(1 −1) ) and D= (_(3 −6) ^(2 −2) ) calculate A= p.D.p^(−1) .

$${let}\:{give}\:{p}=\left(_{\mathrm{1}\:\:\:\:\:\:\:\:\:\:\:−\mathrm{2}} ^{\mathrm{1}\:\:\:\:\:\:\:\:\:\:−\mathrm{1}} \:\:\right)\:\:{and}\:\:{D}=\:\:\left(_{\mathrm{3}\:\:\:\:\:\:\:\:\:\:−\mathrm{6}} ^{\mathrm{2}\:\:\:\:\:\:\:\:−\mathrm{2}} \:\right)\:{calculate} \\ $$$${A}=\:{p}.{D}.{p}^{−\mathrm{1}} \:. \\ $$

Question Number 28256 Answers: 0 Comments: 0

let give A= _( () −1 1 1) ( 1 −1 1) find A^n for n integr. ( 1 1 −1)

$$\left.{let}\:{give}\:\:\:{A}=\:\:_{\:\:\:\:\left(\right.} \:−\mathrm{1}\:\:\:\:\:\:\:\mathrm{1}\:\:\:\:\:\:\mathrm{1}\right) \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\left(\:\:\:\:\:\:\:\:\:\mathrm{1}\:\:\:\:\:\:\:−\mathrm{1}\:\:\:\:\mathrm{1}\right)\:\:\:\:\:\:\:{find}\:\:{A}^{{n}} \:\:{for}\:{n}\:{integr}. \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\left(\:\:\:\:\:\:\:\mathrm{1}\:\:\:\:\:\:\:\:\:\mathrm{1}\:\:\:\:\:\:−\mathrm{1}\right) \\ $$$$\:\:\:\: \\ $$

Question Number 28219 Answers: 0 Comments: 4

Question Number 28211 Answers: 0 Comments: 6

Question Number 28199 Answers: 1 Comments: 0

suppose one of the side of any box that can be carried onto an airplane must be less than 8m. Find the maximum value of such a box if the sum of the three sides can not exceed 46m.

$${suppose}\:{one}\:{of}\:{the}\:{side}\:{of}\:{any} \\ $$$${box}\:{that}\:{can}\:{be}\:{carried}\:{onto}\:{an} \\ $$$${airplane}\:{must}\:{be}\:{less}\:{than}\:\mathrm{8}{m}. \\ $$$${Find}\:{the}\:{maximum}\:{value}\:{of}\:{such} \\ $$$${a}\:{box}\:{if}\:{the}\:{sum}\:{of}\:{the}\:{three}\:{sides} \\ $$$${can}\:{not}\:{exceed}\:\mathrm{46}{m}. \\ $$

Question Number 28198 Answers: 1 Comments: 0

Find the shortest distance from the origin to the curve xy=3

$${Find}\:{the}\:{shortest}\:{distance}\:{from} \\ $$$${the}\:{origin}\:{to}\:{the}\:{curve}\:{xy}=\mathrm{3} \\ $$

Question Number 28190 Answers: 1 Comments: 2

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