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Question Number 48543 Answers: 2 Comments: 0

The number of times the digit 3 will be written when listing the integers from 1 to 1000 is

$$\mathrm{The}\:\mathrm{number}\:\mathrm{of}\:\mathrm{times}\:\mathrm{the}\:\mathrm{digit}\:\mathrm{3}\:\mathrm{will} \\ $$$$\mathrm{be}\:\mathrm{written}\:\mathrm{when}\:\mathrm{listing}\:\mathrm{the}\:\mathrm{integers} \\ $$$$\mathrm{from}\:\mathrm{1}\:\mathrm{to}\:\mathrm{1000}\:\mathrm{is} \\ $$

Question Number 48541 Answers: 1 Comments: 12

Question Number 48540 Answers: 2 Comments: 0

Question Number 48529 Answers: 3 Comments: 0

If cos A+cos B=m and sin A+sin B=n where m, n ≠0, then sin (A+B) is equal to

$$\mathrm{If}\:\:\mathrm{cos}\:{A}+\mathrm{cos}\:{B}={m}\:\mathrm{and}\:\mathrm{sin}\:{A}+\mathrm{sin}\:{B}={n} \\ $$$$\mathrm{where}\:{m},\:{n}\:\neq\mathrm{0},\:\mathrm{then}\:\mathrm{sin}\:\left({A}+{B}\right)\:\mathrm{is}\:\mathrm{equal} \\ $$$$\mathrm{to} \\ $$

Question Number 48528 Answers: 1 Comments: 0

The value of (√3) cot 20°− 4 cos 20° is

$$\mathrm{The}\:\mathrm{value}\:\mathrm{of}\:\sqrt{\mathrm{3}}\:\mathrm{cot}\:\mathrm{20}°−\:\mathrm{4}\:\mathrm{cos}\:\mathrm{20}°\:\:\mathrm{is} \\ $$

Question Number 48527 Answers: 2 Comments: 0

If xy + yz + zx = 1, then tan^(−1) x + tan^(−1) y + tan^(−1) z =

$$\mathrm{If}\:\:{xy}\:+\:{yz}\:+\:{zx}\:=\:\mathrm{1},\:\mathrm{then} \\ $$$$\mathrm{tan}^{−\mathrm{1}} {x}\:+\:\mathrm{tan}^{−\mathrm{1}} {y}\:+\:\mathrm{tan}^{−\mathrm{1}} {z}\:=\: \\ $$

Question Number 48526 Answers: 2 Comments: 0

The maximum and minimum values of a cos 2θ+ b sin 2θ are

$$\mathrm{The}\:\mathrm{maximum}\:\mathrm{and}\:\mathrm{minimum}\:\mathrm{values} \\ $$$$\mathrm{of}\:\:\:{a}\:\mathrm{cos}\:\mathrm{2}\theta+\:{b}\:\mathrm{sin}\:\mathrm{2}\theta\:\:\mathrm{are} \\ $$

Question Number 48525 Answers: 0 Comments: 0

Question Number 48524 Answers: 0 Comments: 0

Prime numbers differing by 2 are called _____.

$$\mathrm{Prime}\:\mathrm{numbers}\:\mathrm{differing}\:\mathrm{by}\:\mathrm{2}\:\mathrm{are} \\ $$$$\mathrm{called}\:\_\_\_\_\_. \\ $$

Question Number 48522 Answers: 1 Comments: 1

Question Number 48517 Answers: 3 Comments: 1

Question Number 48510 Answers: 0 Comments: 2

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.

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

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