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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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