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AllQuestion and Answers: Page 1368

Question Number 69338 Answers: 1 Comments: 1

Question Number 69412 Answers: 1 Comments: 1

Question Number 69411 Answers: 0 Comments: 1

Question Number 69328 Answers: 0 Comments: 1

∫_( 0) ^(π/2) ((φ(x))/(φ(x)+φ((π/2) −x))) dx =

$$\underset{\:\mathrm{0}} {\overset{\pi/\mathrm{2}} {\int}}\:\frac{\phi\left({x}\right)}{\phi\left({x}\right)+\phi\left(\frac{\pi}{\mathrm{2}}\:−{x}\right)}\:{dx}\:= \\ $$

Question Number 69482 Answers: 1 Comments: 1

lim_(n→∞) ((2+cosn)/(4n+sinn)) = ?

$$\underset{{n}\rightarrow\infty} {{lim}}\frac{\mathrm{2}+{cosn}}{\mathrm{4}{n}+{sinn}}\:=\:? \\ $$

Question Number 74637 Answers: 1 Comments: 1

1)calculate f(x)=∫_0 ^1 t^2 (√(x^2 +t^2 ))dt with x>0 2) calculste g(x)=∫_0 ^1 (t^2 /(√(x^2 +t^2 )))dt

$$\left.\mathrm{1}\right){calculate}\:{f}\left({x}\right)=\int_{\mathrm{0}} ^{\mathrm{1}} {t}^{\mathrm{2}} \sqrt{{x}^{\mathrm{2}} +{t}^{\mathrm{2}} }{dt}\:\:{with}\:{x}>\mathrm{0} \\ $$$$\left.\mathrm{2}\right)\:{calculste}\:{g}\left({x}\right)=\int_{\mathrm{0}} ^{\mathrm{1}} \frac{{t}^{\mathrm{2}} }{\sqrt{{x}^{\mathrm{2}} +{t}^{\mathrm{2}} }}{dt} \\ $$

Question Number 69319 Answers: 0 Comments: 4

Show that lim_((x,y)→(0,0)) (3/(x^2 + 2y^2 )) does not exist

$$\mathrm{Show}\:\mathrm{that}\: \\ $$$$\underset{\left({x},{y}\right)\rightarrow\left(\mathrm{0},\mathrm{0}\right)} {\mathrm{lim}}\:\frac{\mathrm{3}}{{x}^{\mathrm{2}} \:+\:\mathrm{2}{y}^{\mathrm{2}} } \\ $$$$\mathrm{does}\:\mathrm{not}\:\mathrm{exist} \\ $$

Question Number 69385 Answers: 2 Comments: 0

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)

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

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