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Question Number 69327 Answers: 1 Comments: 0

∫_( 0) ^1 ((2^(2x+1) − 5^(2x−1) )/(10^x )) dx =

$$\:\underset{\:\mathrm{0}} {\overset{\mathrm{1}} {\int}}\:\:\frac{\mathrm{2}^{\mathrm{2}{x}+\mathrm{1}} −\:\mathrm{5}^{\mathrm{2}{x}−\mathrm{1}} }{\mathrm{10}^{{x}} }\:{dx}\:=\: \\ $$

Question Number 69293 Answers: 0 Comments: 3

Question Number 69297 Answers: 1 Comments: 1

Question Number 69296 Answers: 0 Comments: 6

Question Number 69314 Answers: 1 Comments: 1

if 5 x y 40 are in GP .find x and y

$${if}\:\mathrm{5}\:{x}\:{y}\:\mathrm{40}\:{are}\:{in}\:{GP}\:.{find}\:{x}\:{and}\:{y} \\ $$

Question Number 69272 Answers: 1 Comments: 0

Question Number 69268 Answers: 1 Comments: 0

Question Number 69261 Answers: 1 Comments: 0

find ∫_0 ^1 xtanx dx

$${find}\:\int_{\mathrm{0}} ^{\mathrm{1}} \:{xtanx}\:{dx} \\ $$

Question Number 69257 Answers: 0 Comments: 0

∫_(−2) ^( 4) ∣x∣^(2x^3 ) dx = ?

$$\int_{−\mathrm{2}} ^{\:\:\mathrm{4}} \mid\boldsymbol{{x}}\mid^{\mathrm{2}\boldsymbol{{x}}^{\mathrm{3}} } \boldsymbol{{dx}}\:=\:? \\ $$

Question Number 69255 Answers: 1 Comments: 0

lim_(x→2) (√((x^3 − 4)/(x^2 − 3x + 2)))

$$\underset{{x}\rightarrow\mathrm{2}} {\mathrm{lim}}\:\:\sqrt{\frac{\mathrm{x}^{\mathrm{3}} \:−\:\mathrm{4}}{\mathrm{x}^{\mathrm{2}} \:−\:\mathrm{3x}\:+\:\mathrm{2}}} \\ $$

Question Number 69276 Answers: 0 Comments: 3

f(x)=Σ_(k=1) ^n ∣x+k∣ (1) find the values of x such that f(x) is minumum. (2) fund the roots of f(x)−m=0 as example you can set n=100, m=2500.

$${f}\left({x}\right)=\underset{{k}=\mathrm{1}} {\overset{{n}} {\sum}}\mid{x}+{k}\mid \\ $$$$\left(\mathrm{1}\right)\:{find}\:{the}\:{values}\:{of}\:{x}\:{such}\:{that}\:{f}\left({x}\right)\: \\ $$$${is}\:{minumum}. \\ $$$$\left(\mathrm{2}\right)\:{fund}\:{the}\:{roots}\:{of}\:{f}\left({x}\right)−{m}=\mathrm{0} \\ $$$$ \\ $$$${as}\:{example}\:{you}\:{can}\:{set}\:{n}=\mathrm{100},\:{m}=\mathrm{2500}. \\ $$

Question Number 69247 Answers: 0 Comments: 1

show that c∣a ⇔ −c∣a.

$${show}\:{that}\: \\ $$$$\:{c}\mid{a}\:\Leftrightarrow\:−{c}\mid{a}. \\ $$

Question Number 69246 Answers: 0 Comments: 0

Explicit ∫_0 ^∞ ((Si(ax))/(x+b)) dx with Si(u)=∫_0 ^u ((sinx)/x)dx

$$\:\:\:{Explicit}\:\:\int_{\mathrm{0}} ^{\infty} \:\:\frac{{Si}\left({ax}\right)}{{x}+{b}}\:{dx}\:\:\:{with}\:\:{Si}\left({u}\right)=\int_{\mathrm{0}} ^{{u}} \:\frac{{sinx}}{{x}}{dx} \\ $$

Question Number 69243 Answers: 0 Comments: 0

Question Number 69241 Answers: 0 Comments: 0

Let consider K=∫_0 ^1 (((1−x^a )(1−x^b )(1−x^c ))/((x−1)lnx))dx prove that e^K = (((a+b)!(a+c)!(b+c)!)/(a!b!c!(a+b+c)!))

$$\:{Let}\:{consider}\:{K}=\int_{\mathrm{0}} ^{\mathrm{1}} \:\frac{\left(\mathrm{1}−{x}^{{a}} \right)\left(\mathrm{1}−{x}^{{b}} \right)\left(\mathrm{1}−{x}^{{c}} \right)}{\left({x}−\mathrm{1}\right){lnx}}{dx}\: \\ $$$${prove}\:{that}\: \\ $$$${e}^{{K}} =\:\frac{\left({a}+{b}\right)!\left({a}+{c}\right)!\left({b}+{c}\right)!}{{a}!{b}!{c}!\left({a}+{b}+{c}\right)!}\:\: \\ $$

Question Number 69238 Answers: 1 Comments: 1

Use Residus Theorem to explicit f(a)=Σ_(n=1) ^∞ (((−1)^n sin(na))/n^3 )

$${Use}\:{Residus}\:{Theorem}\:{to}\:{explicit}\: \\ $$$${f}\left({a}\right)=\underset{{n}=\mathrm{1}} {\overset{\infty} {\sum}}\:\frac{\left(−\mathrm{1}\right)^{{n}} {sin}\left({na}\right)}{{n}^{\mathrm{3}} }\:\: \\ $$

Question Number 69236 Answers: 0 Comments: 0

Use Residus theorem to prove that ∀ a>0 Σ_(n=0) ^∞ (1/( n^2 +a^2 )) = (1/2)((π/(ash(πa))) −(1/a^2 )) and Σ_(n=0) ^∞ (((−1)^n )/(n^2 +a^2 )) = (1/2)((( π)/(a.th(πa))) −(1/a^2 )) Assume that we can developp in integer serie the functions f(x)=(x/(shx)) and g(x)=(x/(thx)) Give the DL_2 of f and g around zero Why can′t we use that theorem to explicit f(a)=Σ_(n=0) ^∞ (((−1)^n )/( (2n+1)^2 +a^2 )) ???

$${Use}\:\:{Residus}\:{theorem}\:{to}\:{prove}\:{that}\:\forall\:{a}>\mathrm{0}\:\:\underset{{n}=\mathrm{0}} {\overset{\infty} {\sum}}\:\frac{\mathrm{1}}{\:{n}^{\mathrm{2}} +{a}^{\mathrm{2}} }\:=\:\frac{\mathrm{1}}{\mathrm{2}}\left(\frac{\pi}{{ash}\left(\pi{a}\right)}\:\:\:−\frac{\mathrm{1}}{{a}^{\mathrm{2}} }\right) \\ $$$${and}\:\:\:\:\:\:\underset{{n}=\mathrm{0}} {\overset{\infty} {\sum}}\:\frac{\left(−\mathrm{1}\right)^{{n}} }{{n}^{\mathrm{2}} +{a}^{\mathrm{2}} }\:=\:\frac{\mathrm{1}}{\mathrm{2}}\left(\frac{\:\pi}{{a}.{th}\left(\pi{a}\right)}\:−\frac{\mathrm{1}}{{a}^{\mathrm{2}} }\right) \\ $$$$\:\:\:{Assume}\:{that}\:{we}\:{can}\:{developp}\:{in}\:{integer}\:{serie}\:{the}\:{functions} \\ $$$${f}\left({x}\right)=\frac{{x}}{{shx}}\:\:\:{and}\:{g}\left({x}\right)=\frac{{x}}{{thx}}\: \\ $$$$\:{Give}\:{the}\:{DL}_{\mathrm{2}} \:{of}\:\:{f}\:{and}\:{g}\:{around}\:{zero}\: \\ $$$${Why}\:{can}'{t}\:{we}\:{use}\:{that}\:{theorem}\:{to}\:{explicit} \\ $$$${f}\left({a}\right)=\underset{{n}=\mathrm{0}} {\overset{\infty} {\sum}}\frac{\left(−\mathrm{1}\right)^{{n}} }{\:\left(\mathrm{2}{n}+\mathrm{1}\right)^{\mathrm{2}} +{a}^{\mathrm{2}} }\:\:\:??? \\ $$

Question Number 69233 Answers: 1 Comments: 1

Prove that B=∫_0 ^1 [ln(−lnu)]^2 du = γ^2 + ζ(2)

$${Prove}\:{that}\:\:{B}=\int_{\mathrm{0}} ^{\mathrm{1}} \:\:\:\left[{ln}\left(−{lnu}\right)\right]^{\mathrm{2}} \:{du}\:=\:\gamma^{\mathrm{2}} +\:\zeta\left(\mathrm{2}\right)\:\: \\ $$

Question Number 69231 Answers: 0 Comments: 1

Prove that Σ_(p=0) ^∞ (((−1)^p )/(4p+1)) = ((π−argcoth((√2) ))/(4(√2))) and Σ_(p=0) ^(∞ ) (((−1)^p )/(4p+3)) = ((π+argcoth((√2) ))/(4(√2) ))

$${Prove}\:{that}\:\:\underset{{p}=\mathrm{0}} {\overset{\infty} {\sum}}\:\:\frac{\left(−\mathrm{1}\right)^{{p}} }{\mathrm{4}{p}+\mathrm{1}}\:=\:\frac{\pi−{argcoth}\left(\sqrt{\mathrm{2}}\:\right)}{\mathrm{4}\sqrt{\mathrm{2}}}\:\:{and} \\ $$$$\underset{{p}=\mathrm{0}} {\overset{\infty\:} {\sum}}\:\frac{\left(−\mathrm{1}\right)^{{p}} }{\mathrm{4}{p}+\mathrm{3}}\:=\:\frac{\pi+{argcoth}\left(\sqrt{\mathrm{2}}\:\right)}{\mathrm{4}\sqrt{\mathrm{2}}\:}\:\:\: \\ $$

Question Number 69230 Answers: 0 Comments: 0

Question Number 69229 Answers: 0 Comments: 0

a, b, c ∈ nonnegative real numbers (√(a^2 + b^2 + 1)) + (√(b^2 + c^2 + 1)) + (√(c^2 + a^2 + 1)) ≥ 2 + (√(2(a^2 + b^2 + c^2 ) + 1)) Find all triplets (a, b, c) so that inequality above hold .

$${a},\:{b},\:{c}\:\:\in\:\:{nonnegative}\:\:{real}\:\:{numbers} \\ $$$$\sqrt{{a}^{\mathrm{2}} \:+\:{b}^{\mathrm{2}} \:+\:\mathrm{1}}\:+\:\sqrt{{b}^{\mathrm{2}} \:+\:{c}^{\mathrm{2}} \:+\:\mathrm{1}}\:+\:\sqrt{{c}^{\mathrm{2}} \:+\:{a}^{\mathrm{2}} \:+\:\mathrm{1}}\:\:\geqslant\:\:\mathrm{2}\:+\:\sqrt{\mathrm{2}\left({a}^{\mathrm{2}} \:+\:{b}^{\mathrm{2}} \:+\:{c}^{\mathrm{2}} \right)\:+\:\mathrm{1}} \\ $$$${Find}\:\:{all}\:\:{triplets}\:\left({a},\:{b},\:{c}\right)\:\:{so}\:\:{that}\:\:{inequality}\:\:{above}\:\:{hold}\:. \\ $$

Question Number 69226 Answers: 0 Comments: 0

Question Number 69222 Answers: 0 Comments: 1

Question Number 69212 Answers: 1 Comments: 0

use limit comparison test to determine the series converge or diverge Σ_(k=1) ^∞ ln(1+(1/k^2 ))

$${use}\:{limit}\:{comparison}\:{test}\:{to}\:{determine} \\ $$$${the}\:{series}\:{converge}\:{or}\:{diverge} \\ $$$$\underset{{k}=\mathrm{1}} {\overset{\infty} {\sum}}{ln}\left(\mathrm{1}+\frac{\mathrm{1}}{{k}^{\mathrm{2}} }\right) \\ $$

Question Number 69207 Answers: 1 Comments: 3

help please. A river is 5m wide and flows at 3.0ms^(−1) . A man can swim at 2.0ms^(−1) in still water. if he sets off at an angle of 90° to the bank calculate a) the mans time and velocity b) his distance downstream from the starting point till when he reaches the other side of the river bank c) the actual distance he swims through the water.

$${help}\:{please}. \\ $$$$ \\ $$$${A}\:{river}\:{is}\:\mathrm{5}{m}\:{wide}\:{and}\:{flows}\:{at}\:\mathrm{3}.\mathrm{0}{ms}^{−\mathrm{1}} .\:{A}\:{man}\:{can}\:{swim}\:{at}\:\mathrm{2}.\mathrm{0}{ms}^{−\mathrm{1}} \\ $$$${in}\:{still}\:{water}.\:{if}\:{he}\:{sets}\:{off}\:{at}\:{an}\:{angle}\:{of}\:\mathrm{90}°\:{to}\:{the}\:{bank} \\ $$$${calculate} \\ $$$$\left.{a}\right)\:{the}\:{mans}\:{time}\:{and}\:{velocity} \\ $$$$\left.{b}\right)\:{his}\:{distance}\:{downstream}\:{from}\:{the}\:{starting}\:{point}\:{till} \\ $$$${when}\:{he}\:{reaches}\:{the}\:{other}\:{side}\:{of}\:{the}\:{river}\:{bank} \\ $$$$\left.{c}\right)\:{the}\:{actual}\:{distance}\:{he}\:{swims}\:{through}\:{the}\:{water}. \\ $$

Question Number 69211 Answers: 0 Comments: 0

To conserve rice from a cooperative there is a foil of area A=24cm^2 . The cooperatives want to build a barn in the shape of a parallelopiped square rectangle without cover. What should be the symmetrical dimensions for the volume to be maximum?

$${To}\:{conserve}\:{rice}\:{from}\:{a}\:{cooperative}\:{there}\:{is} \\ $$$${a}\:{foil}\:{of}\:{area}\:{A}=\mathrm{24}{cm}^{\mathrm{2}} .\:{The}\:{cooperatives} \\ $$$${want}\:{to}\:{build}\:{a}\:{barn}\:{in}\:{the}\:{shape}\:{of}\:{a}\:{parallelopiped} \\ $$$${square}\:{rectangle}\:{without}\:{cover}. \\ $$$${What}\:{should}\:{be}\:{the}\:{symmetrical}\:{dimensions}\:{for} \\ $$$${the}\:{volume}\:{to}\:{be}\:{maximum}? \\ $$

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