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Question Number 65911 Answers: 0 Comments: 1

Question Number 65901 Answers: 1 Comments: 0

Question Number 65884 Answers: 0 Comments: 3

Question Number 65878 Answers: 0 Comments: 5

Calculate lim_(a−>∞) ∫_0 ^∞ (dx/(1+x^a ))

$$\:\:{Calculate}\:\:\:{lim}_{{a}−>\infty} \:\:\int_{\mathrm{0}} ^{\infty} \:\:\frac{{dx}}{\mathrm{1}+{x}^{{a}} }\: \\ $$

Question Number 65871 Answers: 0 Comments: 0

To: sirAjfour,sir:mrW1, sir:MJS by considering my comment on: Q#14535 and sir Ajfour and mrW1 answer′s to Q#62839,I think now we can solve Q#14535 .if you have time please try it.I know there is a relation between this questions.but can′t find it. kindly try it.thanks alot sir.

$$\mathrm{To}:\:\mathrm{sirAjfour},\mathrm{sir}:\mathrm{mrW1},\:\mathrm{sir}:\mathrm{MJS} \\ $$$$\mathrm{by}\:\mathrm{considering}\:\mathrm{my}\:\mathrm{comment}\:\mathrm{on}: \\ $$$$\mathrm{Q}#\mathrm{14535}\:\mathrm{and}\:\mathrm{sir}\:\mathrm{Ajfour}\:\mathrm{and}\:\mathrm{mrW1} \\ $$$$\mathrm{answer}'\mathrm{s}\:\mathrm{to}\:\mathrm{Q}#\mathrm{62839},\mathrm{I}\:\mathrm{think}\:\mathrm{now}\:\mathrm{we} \\ $$$$\mathrm{can}\:\mathrm{solve}\:\mathrm{Q}#\mathrm{14535}\:.\mathrm{if}\:\mathrm{you}\:\mathrm{have}\:\mathrm{time} \\ $$$$\mathrm{please}\:\mathrm{try}\:\mathrm{it}.\mathrm{I}\:\mathrm{know}\:\mathrm{there}\:\mathrm{is}\:\mathrm{a}\:\mathrm{relation} \\ $$$$\mathrm{between}\:\mathrm{this}\:\mathrm{questions}.\mathrm{but}\:\mathrm{can}'\mathrm{t}\:\mathrm{find}\:\mathrm{it}. \\ $$$$\mathrm{kindly}\:\mathrm{try}\:\mathrm{it}.\mathrm{thanks}\:\mathrm{alot}\:\mathrm{sir}. \\ $$

Question Number 65869 Answers: 1 Comments: 1

∫((2x^2 −3x+4)/(4x^3 +5)) dx

$$\int\frac{\mathrm{2}{x}^{\mathrm{2}} −\mathrm{3}{x}+\mathrm{4}}{\mathrm{4}{x}^{\mathrm{3}} +\mathrm{5}}\:{dx} \\ $$

Question Number 65868 Answers: 0 Comments: 2

Question Number 65866 Answers: 0 Comments: 5

Question Number 65859 Answers: 1 Comments: 0

x^4 +5x^2 +20x+104=0 solve for x.

$${x}^{\mathrm{4}} +\mathrm{5}{x}^{\mathrm{2}} +\mathrm{20}{x}+\mathrm{104}=\mathrm{0} \\ $$$${solve}\:{for}\:{x}. \\ $$

Question Number 65858 Answers: 0 Comments: 3

Question Number 65853 Answers: 1 Comments: 0

x^4 −23x^2 +18x+40=0 solve for x.

$${x}^{\mathrm{4}} −\mathrm{23}{x}^{\mathrm{2}} +\mathrm{18}{x}+\mathrm{40}=\mathrm{0} \\ $$$${solve}\:{for}\:{x}. \\ $$

Question Number 65841 Answers: 1 Comments: 1

(d/dx)(((tan^2 x)/(1 + cos x))) =?

$$\:\frac{{d}}{{dx}}\left(\frac{{tan}\:^{\mathrm{2}} {x}}{\mathrm{1}\:+\:{cos}\:{x}}\right)\:=? \\ $$

Question Number 65837 Answers: 0 Comments: 4

1) calculate ∫_(−∞) ^∞ (dx/(1+ix)) and ∫_(−∞) ^∞ (dx/(1−ix)) 2)deduce the value of ∫_(−∞) ^∞ (dx/(1+x^2 )) 3)calculate ∫_(−∞) ^∞ (dx/(1+ix^2 )) and ∫_(−∞) ^∞ (dx/(1−ix^2 )) 4)deduce the value of ∫_(−∞) ^∞ (dx/(1+x^4 ))

$$\left.\mathrm{1}\right)\:{calculate}\:\int_{−\infty} ^{\infty} \:\frac{{dx}}{\mathrm{1}+{ix}}\:\:{and}\:\int_{−\infty} ^{\infty} \:\:\frac{{dx}}{\mathrm{1}−{ix}} \\ $$$$\left.\mathrm{2}\right){deduce}\:{the}\:{value}\:{of}\:\int_{−\infty} ^{\infty} \:\frac{{dx}}{\mathrm{1}+{x}^{\mathrm{2}} } \\ $$$$\left.\mathrm{3}\right){calculate}\:\int_{−\infty} ^{\infty} \:\:\frac{{dx}}{\mathrm{1}+{ix}^{\mathrm{2}} }\:\:{and}\:\int_{−\infty} ^{\infty} \:\frac{{dx}}{\mathrm{1}−{ix}^{\mathrm{2}} } \\ $$$$\left.\mathrm{4}\right){deduce}\:{the}\:{value}\:{of}\:\int_{−\infty} ^{\infty} \:\frac{{dx}}{\mathrm{1}+{x}^{\mathrm{4}} } \\ $$

Question Number 65834 Answers: 0 Comments: 1

∀ x, y >0 B(x,y)=∫_0 ^1 t^(x−1) (1−t)^(y−1) dt Γ(x)=∫_0 ^∞ t^(x−1) e^(−t) dt 1) show that ∀ x>0 Γ(x+1)=xΓ(x) and lim_(n−>∞) ((x(x+1)......(x+n))/(n^x n!))=(1/(Γ(x))) and deduce that lim_(n−>∞) ((Γ(x+n))/(n^x Γ(n)))=1 b) Prove that if a function f satisfies f(x+1)=xf(x) et lim_(n−>∞) ((f(x+n))/(n^x f(n)))=1 then ∀ x>0 f(x)= f(1)Γ(x) 3) Show that B(x+1, y)=(x/(x+y))B(x,y) B(1,x)=(1/x) 2) Now let consider ∀ y f(x)=((B(x,y)Γ(x+y))/(Γ(y))) Show that f verify the same property as Γ ( just the both proved up ) 3) Deduce that ∀ x,y>0 B(x,y)=((Γ(x)Γ(y))/(Γ(x+y)))

$$\:\:\:\forall\:\:{x},\:{y}\:\:>\mathrm{0}\:\:\:\:{B}\left({x},{y}\right)=\int_{\mathrm{0}} ^{\mathrm{1}} \:{t}^{{x}−\mathrm{1}} \left(\mathrm{1}−{t}\right)^{{y}−\mathrm{1}} {dt}\:\:\:\:\:\:\Gamma\left({x}\right)=\int_{\mathrm{0}} ^{\infty} \:{t}^{{x}−\mathrm{1}} \:{e}^{−{t}} {dt} \\ $$$$\left.\mathrm{1}\right)\:{show}\:{that}\:\:\forall\:{x}>\mathrm{0}\:\:\:\:\Gamma\left({x}+\mathrm{1}\right)={x}\Gamma\left({x}\right)\:\:\:\:{and}\:\:{lim}_{{n}−>\infty} \:\frac{{x}\left({x}+\mathrm{1}\right)......\left({x}+{n}\right)}{{n}^{{x}} {n}!}=\frac{\mathrm{1}}{\Gamma\left({x}\right)} \\ $$$${and}\:{deduce}\:{that}\:\:{lim}_{{n}−>\infty} \:\frac{\Gamma\left({x}+{n}\right)}{{n}^{{x}} \:\Gamma\left({n}\right)}=\mathrm{1} \\ $$$$\left.{b}\right)\:{Prove}\:{that}\:{if}\:\:{a}\:\:{function}\:{f}\:{satisfies}\:\:{f}\left({x}+\mathrm{1}\right)={xf}\left({x}\right)\:\:{et}\:\:\:{lim}_{{n}−>\infty} \:\:\frac{{f}\left({x}+{n}\right)}{{n}^{{x}} \:{f}\left({n}\right)}=\mathrm{1}\:\:{then}\:\forall\:{x}>\mathrm{0}\:\:{f}\left({x}\right)=\:{f}\left(\mathrm{1}\right)\Gamma\left({x}\right) \\ $$$$\left.\mathrm{3}\right)\:\:{Show}\:{that}\:\:{B}\left({x}+\mathrm{1},\:{y}\right)=\frac{{x}}{{x}+{y}}{B}\left({x},{y}\right)\:\:\:\:\:\:{B}\left(\mathrm{1},{x}\right)=\frac{\mathrm{1}}{{x}}\: \\ $$$$\left.\mathrm{2}\right)\:{Now}\:{let}\:{consider}\:\forall\:{y}\:\:\:{f}\left({x}\right)=\frac{{B}\left({x},{y}\right)\Gamma\left({x}+{y}\right)}{\Gamma\left({y}\right)}\: \\ $$$${Show}\:{that}\:\:{f}\:\:{verify}\:\:{the}\:\:{same}\:{property}\:{as}\:\Gamma\:\:\left(\:{just}\:{the}\:{both}\:{proved}\:{up}\:\right)\: \\ $$$$\left.\mathrm{3}\right)\:{Deduce}\:\:{that}\:\forall\:{x},{y}>\mathrm{0}\:\:\:{B}\left({x},{y}\right)=\frac{\Gamma\left({x}\right)\Gamma\left({y}\right)}{\Gamma\left({x}+{y}\right)} \\ $$

Question Number 65830 Answers: 1 Comments: 3

Question Number 65828 Answers: 0 Comments: 1

Let go toward a rational order of derivation Part 1 : What′s that special factor Let n , p and k three integer different of zero We state J_(n,k) (p)=∫_0 ^1 (1−x^n )^(p+(k/n)) dx and C_n (p)=Π_(k=0) ^(n−1) J_(n,k) (p) 1) a) Calculate C_1 (p) b) Prove that J_(n,k) (p)=(1/n)B((1/n),p+1+(k/n) ) and explicit C_n (p)in terms of n and p 2) Deduce that ∀ n>0 there exist a real a_n such as (na_n )^n C_n (p)= (1/(p+1)) 3) Study the convergence of the result suite (a_n )_n .Then show that lim_(n−>∞) na_n =1 Part 2: the rational order of derivation Let f ∈ C^1 (R,R) . We consider I_(1/n) (f) a function defined on R_+ by I_(1/n) (f)(x)= a_n ∫_0 ^x ((f(t))/((x−t)^(1−(1/n)) ))dt and D_(1/n) (f) = (I_(1/n) (f))^((1)) 1) a _ Prove that I_((1/n) ) (f)(x)= na_n x^(1/n) ∫_0 ^1 f(x(1−v^n ))dv then find D_(1/2) (t) b) Show that ∀ f∈C^1 (R,R) ∀ x∈R_(+ ) D_(1/n) (f)(x)= I_(1/n) (f)(x) + ((f(0))/((πx)^(1−(1/n)) )) 2)∀ p integer and k∈{0,...,n−1} explicit I_(1/n) (t^(p+(k/n)) ) in term of I_(n,k) (p) b) Prove that for polynomial function f the n− th composition I_(1/n) ._ ....I_(1/n) (f)(x)=∫_0 ^x f(t)dt , c) Deduce that ∀ f polynomial the function g =f −f(0) verify D_(1/n) ......D_(1/n) (g)(x) = g(x) 3) Widen that two formulas to all function that can be developp into integer serie 4) Try to find the relation between D_(1/n) .I_(1/n) (f) , I_(1/n) .D_(1/n) (f), and f 4) Show that ∀ x∈R_+ lim_(n−>∞) I_(1/n) (f)(x)= ∫_0 ^x f(t)dt pour g=f−f(0) lim_(n−>∞) D_(1/n) (g)(x)= g(x) conclusion the derivative of the function I_α (f) defined on R_+ by I_α (f)(x)= a_n ∫_0 ^x f(t)(x−t)^((1/n)−1) dt is called the derivative of order α

Question Number 65827 Answers: 0 Comments: 0

Prove that ∫_0 ^1 (∫_(1/6) ^(5/6) (dv/((1−^v (√u) )^v )))du=ln2−2ln((√3)−1)

$${Prove}\:{that}\:\: \\ $$$$\int_{\mathrm{0}} ^{\mathrm{1}} \left(\int_{\frac{\mathrm{1}}{\mathrm{6}}} ^{\frac{\mathrm{5}}{\mathrm{6}}} \:\:\frac{{dv}}{\left(\mathrm{1}−\:^{{v}} \sqrt{{u}}\:\right)^{{v}} }\right){du}={ln}\mathrm{2}−\mathrm{2}{ln}\left(\sqrt{\mathrm{3}}−\mathrm{1}\right) \\ $$

Question Number 65825 Answers: 0 Comments: 0

let consider two real numbers p and such as p^2 −q^2 =pq Prove that J= ∫_0 ^∞ (dv/(^q (√((1+^q (√(v^p )) )^p ))))= 1

$${let}\:{consider}\:\:{two}\:{real}\:{numbers}\:{p}\:{and}\:{such}\:{as}\:{p}^{\mathrm{2}} −{q}^{\mathrm{2}} ={pq} \\ $$$${Prove}\:{that} \\ $$$$\:\:\:{J}=\:\int_{\mathrm{0}} ^{\infty} \:\:\:\:\frac{{dv}}{\:^{{q}} \sqrt{\left(\mathrm{1}+\:^{{q}} \sqrt{{v}^{{p}} \:}\:\right)^{{p}} }}=\:\mathrm{1} \\ $$$$ \\ $$

Question Number 65805 Answers: 2 Comments: 1

Prove that I_n =∫_0 ^(π/2) (dt/(1+(tant)^n )) does not depend of the term n deduces that ∫_0 ^∞ (dx/((x^(2035) +1)(x^2 +1)))=(π/4)

$$\:\:{Prove}\:{that}\:\:{I}_{{n}} =\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \:\:\:\frac{{dt}}{\mathrm{1}+\left({tant}\right)^{{n}} }\:\:{does}\:{not}\:{depend}\:{of}\:{the}\:{term}\:{n} \\ $$$${deduces}\:{that} \\ $$$$\int_{\mathrm{0}} ^{\infty} \:\:\frac{{dx}}{\left({x}^{\mathrm{2035}} +\mathrm{1}\right)\left({x}^{\mathrm{2}} +\mathrm{1}\right)}=\frac{\pi}{\mathrm{4}} \\ $$

Question Number 65797 Answers: 1 Comments: 0

find the constant a,b and c so that the direction derivative of Φ=axy^2 +byz+cz^2 x^3 at (1,2,−1) has a maximum of magnitude 64 jn a direction parallel to the z axis.

$${find}\:{the}\:{constant}\:\:{a},{b}\:{and}\:\:{c}\:\:{so} \\ $$$${that}\:{the}\:{direction}\:{derivative}\:{of} \\ $$$$\Phi={axy}^{\mathrm{2}} +{byz}+{cz}^{\mathrm{2}} {x}^{\mathrm{3}} \:{at}\:\left(\mathrm{1},\mathrm{2},−\mathrm{1}\right) \\ $$$${has}\:{a}\:{maximum}\:{of}\:{magnitude} \\ $$$$\mathrm{64}\:{jn}\:{a}\:{direction}\:{parallel}\:{to}\:{the} \\ $$$${z}\:{axis}. \\ $$

Question Number 65788 Answers: 0 Comments: 0

Explicit f(a.b.c)=∫_0 ^(π/2) ((sec(x−a))/(b.cosx + c.sinx)) dx

$${Explicit}\:\:\:{f}\left({a}.{b}.{c}\right)=\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \:\:\frac{{sec}\left({x}−{a}\right)}{{b}.{cosx}\:+\:{c}.{sinx}}\:{dx} \\ $$$$ \\ $$

Question Number 65786 Answers: 0 Comments: 0

Shows that ∣Γ(1+ix)∣^2 =(π/(xsinh(πx))) with Γ(z)=∫_0_ ^∞ t^(z−1) e^(−t) dt Then Prove that ∫_0 ^∞ ∣Γ(1+ix)∣^2 dx =(π/4)

$$\:{Shows}\:{that}\:\:\mid\Gamma\left(\mathrm{1}+{ix}\right)\mid^{\mathrm{2}} =\frac{\pi}{{xsinh}\left(\pi{x}\right)}\:\:\:\:\:\:{with}\:\Gamma\left({z}\right)=\int_{\mathrm{0}_{} } ^{\infty} \:{t}^{{z}−\mathrm{1}} {e}^{−{t}} {dt} \\ $$$${Then}\:{Prove}\:{that}\:\:\int_{\mathrm{0}} ^{\infty} \:\mid\Gamma\left(\mathrm{1}+{ix}\right)\mid^{\mathrm{2}} \:{dx}\:=\frac{\pi}{\mathrm{4}} \\ $$

Question Number 65782 Answers: 0 Comments: 4

Evaluate ∫_0 ^2 (3x^2 −2x + 4)^7 dx hence show that (d/dx)(coshx) = sinh x

$$\:{Evaluate}\:\int_{\mathrm{0}} ^{\mathrm{2}} \left(\mathrm{3}{x}^{\mathrm{2}} −\mathrm{2}{x}\:+\:\mathrm{4}\right)^{\mathrm{7}} {dx} \\ $$$${hence}\:{show}\:{that}\:\:\frac{{d}}{{dx}}\left({coshx}\right)\:=\:{sinh}\:{x} \\ $$

Question Number 65781 Answers: 1 Comments: 0

If xyz ≠ 0 and x+y+z=0 a=10^z b=10^y c=10^x then a^(((1/y)+(1/z))) . b^(((1/z)+(1/x))) .c^(((1/x)+(1/y))) =... a. 0.001 b. 0.01 c. 0.1 d. 1 e. 10