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Question Number 41347 Answers: 1 Comments: 0

Question Number 41346 Answers: 0 Comments: 6

calculate ∫∫_([0,1]^2 ) cos(x^2 +y^2 )dxdy .

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

Question Number 41345 Answers: 0 Comments: 2

let u_n = Σ_(k=1) ^n (1/k) −ln(n) 1) prove that (u_n )is convergent 2) let γ =lim_(n→+∞) u_n prove that 0<γ<1

$${let}\:{u}_{{n}} =\:\sum_{{k}=\mathrm{1}} ^{{n}} \:\frac{\mathrm{1}}{{k}}\:−{ln}\left({n}\right) \\ $$$$\left.\mathrm{1}\right)\:{prove}\:{that}\:\left({u}_{{n}} \right){is}\:{convergent} \\ $$$$\left.\mathrm{2}\right)\:{let}\:\gamma\:={lim}_{{n}\rightarrow+\infty} {u}_{{n}} \:\:\:{prove}\:{that}\:\mathrm{0}<\gamma<\mathrm{1}\:\: \\ $$

Question Number 41343 Answers: 1 Comments: 1

calculate ∫∫_D (x^2 −y^2 )dxdy with D = [−1,1]^2

$${calculate}\:\:\:\:\int\int_{{D}} \:\left({x}^{\mathrm{2}} −{y}^{\mathrm{2}} \right){dxdy}\:\:{with} \\ $$$${D}\:=\:\left[−\mathrm{1},\mathrm{1}\right]^{\mathrm{2}} \\ $$

Question Number 41342 Answers: 0 Comments: 0

find the value of Σ_(n=1) ^∞ (1/(n^2 (n+1)^2 (n+2)^2 ))

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

Question Number 41361 Answers: 1 Comments: 0

The half life of radium-226 is 1620 years.Calculate (a)the decay constant (b)the time it takes 60% of a given sample to decay, and (c)the initial activity of 1gm of pure radium-226

$${The}\:{half}\:{life}\:{of}\:{radium}-\mathrm{226}\:{is}\:\mathrm{1620} \\ $$$${years}.{Calculate}\:\left({a}\right){the}\:{decay}\:{constant} \\ $$$$\left({b}\right){the}\:{time}\:{it}\:{takes}\:\mathrm{60\%}\:{of}\:{a}\:{given} \\ $$$${sample}\:{to}\:{decay},\:{and}\:\left({c}\right){the}\:{initial} \\ $$$${activity}\:{of}\:\mathrm{1}{gm}\:{of}\:{pure}\:{radium}-\mathrm{226} \\ $$

Question Number 41332 Answers: 1 Comments: 3

In Matrices, Is (A^(−1) )B = B(A^(−1) ) ?

$$\mathrm{In}\:\mathrm{Matrices},\: \\ $$$$\mathrm{Is}\:\left(\mathrm{A}^{−\mathrm{1}} \right)\mathrm{B}\:=\:\mathrm{B}\left(\mathrm{A}^{−\mathrm{1}} \right)\:? \\ $$

Question Number 41327 Answers: 0 Comments: 0

Derive the sum of an Harmonic Progression

$${Derive}\:{the}\:{sum}\:{of}\:{an}\:{Harmonic}\:{Progression} \\ $$

Question Number 41326 Answers: 0 Comments: 0

Evaluate ∫_0 ^(π/2) x^3 sec^5 x dx

$${Evaluate}\:\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} {x}^{\mathrm{3}} {sec}^{\mathrm{5}} {x}\:{dx} \\ $$

Question Number 41324 Answers: 1 Comments: 2

Question Number 41321 Answers: 1 Comments: 1

If A is a square matrix of order 3, then ∣(A−A^T )^(2011) ∣ = ?

$$\mathrm{If}\:\mathrm{A}\:\mathrm{is}\:\mathrm{a}\:\mathrm{square}\:\mathrm{matrix}\:\mathrm{of}\:\mathrm{order}\:\mathrm{3},\:\mathrm{then} \\ $$$$\:\mid\left(\mathrm{A}−\mathrm{A}^{\mathrm{T}} \right)^{\mathrm{2011}} \mid\:=\:? \\ $$

Question Number 41305 Answers: 1 Comments: 1

[(1,2,3),(3,4,5),(5,6,7) ] find determinant of given matrix?

$$\begin{bmatrix}{\mathrm{1}}&{\mathrm{2}}&{\mathrm{3}}\\{\mathrm{3}}&{\mathrm{4}}&{\mathrm{5}}\\{\mathrm{5}}&{\mathrm{6}}&{\mathrm{7}}\end{bmatrix} \\ $$$${find}\:{determinant}\:{of}\:{given}\:{matrix}? \\ $$

Question Number 41302 Answers: 1 Comments: 1

calculate ∫_1 ^(+∞) (dx/(x^2 (√(x^2 +x+1))))

$${calculate}\:\:\int_{\mathrm{1}} ^{+\infty} \:\:\frac{{dx}}{{x}^{\mathrm{2}} \sqrt{{x}^{\mathrm{2}} +{x}+\mathrm{1}}} \\ $$

Question Number 41301 Answers: 1 Comments: 4

let f(x)=∫_0 ^∞ e^(−ax) ln(1+e^(−bx) )dx with a>0 and b>0 1) calculate (∂f/∂a)(x) 2) calculate (∂f/∂b)(x) 3)find the value of ∫_0 ^∞ e^(−2x) ln(1+e^(−x) )dx and ∫_0 ^∞ e^(−x) ln(1+e^(−2x) )dx .

$${let}\:{f}\left({x}\right)=\int_{\mathrm{0}} ^{\infty} \:\:{e}^{−{ax}} {ln}\left(\mathrm{1}+{e}^{−{bx}} \right){dx}\:{with}\:{a}>\mathrm{0}\:{and} \\ $$$${b}>\mathrm{0} \\ $$$$\left.\mathrm{1}\right)\:{calculate}\:\frac{\partial{f}}{\partial{a}}\left({x}\right) \\ $$$$\left.\mathrm{2}\right)\:{calculate}\:\frac{\partial{f}}{\partial{b}}\left({x}\right) \\ $$$$\left.\mathrm{3}\right){find}\:{the}\:{value}\:{of}\:\int_{\mathrm{0}} ^{\infty} \:{e}^{−\mathrm{2}{x}} {ln}\left(\mathrm{1}+{e}^{−{x}} \right){dx}\:{and} \\ $$$$\int_{\mathrm{0}} ^{\infty} \:\:{e}^{−{x}} {ln}\left(\mathrm{1}+{e}^{−\mathrm{2}{x}} \right){dx}\:. \\ $$

Question Number 41292 Answers: 1 Comments: 0

Question Number 41291 Answers: 2 Comments: 0

Let ABCD be a parallelogram whose diagonals intersect at P and ley O be the origin, then OA^(→) +OB^(→) +OC^(→) +OD^(→) equals

$$\mathrm{Let}\:{ABCD}\:\mathrm{be}\:\mathrm{a}\:\mathrm{parallelogram}\:\mathrm{whose} \\ $$$$\mathrm{diagonals}\:\mathrm{intersect}\:\mathrm{at}\:{P}\:\mathrm{and}\:\mathrm{ley}\:{O}\:\mathrm{be} \\ $$$$\mathrm{the}\:\mathrm{origin},\:\mathrm{then}\:\overset{\rightarrow} {{OA}}+\overset{\rightarrow} {{OB}}+\overset{\rightarrow} {{OC}}+\overset{\rightarrow} {{OD}}\: \\ $$$$\mathrm{equals} \\ $$

Question Number 41290 Answers: 1 Comments: 0

An electric pole PN is such that PN=12cm where N is the top of the pole and P the base .At a given moment of the day the shadow of the pole PN′ = PN. find a) the length NN′ b) the bearing of P from N.

$${An}\:{electric}\:{pole}\:{PN}\:{is}\:{such}\:{that}\:{PN}=\mathrm{12}{cm}\:{where}\:{N}\:{is}\:{the}\:{top}\:{of}\:{the}\:{pole}\:{and}\:{P}\:{the}\:{base} \\ $$$$.{At}\:{a}\:{given}\:{moment}\:{of}\:{the}\:{day}\:{the}\:\boldsymbol{{shadow}}\:\boldsymbol{{of}}\:\boldsymbol{{the}}\:\boldsymbol{{pole}}\:{PN}'\:=\:{PN}.\:{find}\: \\ $$$$\left.{a}\right)\:{the}\:{length}\:{NN}' \\ $$$$\left.{b}\right)\:{the}\:{bearing}\:{of}\:{P}\:{from}\:{N}. \\ $$

Question Number 41288 Answers: 1 Comments: 0

by using the knowledge of sequence and series show that the compound interest is given by An=P(1+((RT)/(100)))^n

Question Number 41280 Answers: 1 Comments: 1

find f(x) = ∫_0 ^1 arctan(xt^2 )dt

$${find}\:\:{f}\left({x}\right)\:=\:\int_{\mathrm{0}} ^{\mathrm{1}} \:\:{arctan}\left({xt}^{\mathrm{2}} \right){dt} \\ $$

Question Number 41279 Answers: 0 Comments: 1

let f(x)=∫_0 ^∞ arctan(xt^2 )dt . find a explicite form of f^′ (x)

$${let}\:{f}\left({x}\right)=\int_{\mathrm{0}} ^{\infty} \:\:\:{arctan}\left({xt}^{\mathrm{2}} \right){dt}\:. \\ $$$${find}\:\:{a}\:{explicite}\:{form}\:{of}\:{f}^{'} \left({x}\right) \\ $$

Question Number 41273 Answers: 0 Comments: 2

find f(x)=∫_0 ^(+∞) arctan(xt^2 )dt with x fromR .

$${find}\:\:{f}\left({x}\right)=\int_{\mathrm{0}} ^{+\infty} \:{arctan}\left({xt}^{\mathrm{2}} \right){dt}\:\:{with}\:{x}\:{fromR}\:. \\ $$