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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

$$\:{Let}\:{go}\:{toward}\:{a}\:{rational}\:{order}\:{of}\:{derivation} \\ $$$$ \\ $$$${Part}\:\mathrm{1}\::\:\:{What}'{s}\:{that}\:{special}\:{factor}\:\: \\ $$$${Let}\:{n}\:,\:{p}\:{and}\:{k}\:{three}\:{integer}\:\:{different}\:{of}\:{zero} \\ $$$${We}\:\:{state}\:{J}_{{n},{k}} \left({p}\right)=\int_{\mathrm{0}} ^{\mathrm{1}} \:\:\left(\mathrm{1}−{x}^{{n}} \right)^{{p}+\frac{{k}}{{n}}} {dx}\:\:\:{and}\:\:{C}_{{n}} \left({p}\right)=\underset{{k}=\mathrm{0}} {\overset{{n}−\mathrm{1}} {\prod}}\:{J}_{{n},{k}} \left({p}\right) \\ $$$$\left.\mathrm{1}\left.\right)\:{a}\right)\:{Calculate}\:{C}_{\mathrm{1}} \left({p}\right)\:\: \\ $$$$\left.\:\:\:\:{b}\right)\:{Prove}\:{that}\:\:{J}_{{n},{k}} \left({p}\right)=\frac{\mathrm{1}}{{n}}{B}\left(\frac{\mathrm{1}}{{n}},{p}+\mathrm{1}+\frac{{k}}{{n}}\:\right)\:\:\:{and}\:\:{explicit}\:\:{C}_{{n}} \left({p}\right){in}\:{terms}\:{of}\:\:{n}\:{and}\:{p}\: \\ $$$$\left.\mathrm{2}\right)\:{Deduce}\:{that}\:\forall\:{n}>\mathrm{0}\:\:\:\:{there}\:{exist}\:{a}\:{real}\:{a}_{{n}} \:{such}\:{as}\:\:\left({na}_{{n}} \right)^{{n}} {C}_{{n}} \left({p}\right)=\:\frac{\mathrm{1}}{{p}+\mathrm{1}}\: \\ $$$$\left.\mathrm{3}\right)\:{Study}\:{the}\:{convergence}\:{of}\:{the}\:{result}\:{suite}\:\left({a}_{{n}} \right)_{{n}} \:\:\:.{Then}\:{show}\:{that}\:{lim}_{{n}−>\infty} \:{na}_{{n}} \:=\mathrm{1} \\ $$$${Part}\:\mathrm{2}:\:\:{the}\:{rational}\:{order}\:{of}\:{derivation} \\ $$$${Let}\:\:{f}\:\in\:{C}^{\mathrm{1}} \left(\mathbb{R},\mathbb{R}\right)\:.\:{We}\:\:{consider}\:{I}_{\frac{\mathrm{1}}{{n}}} \left({f}\right)\:{a}\:{function}\:{defined}\:{on}\:\mathbb{R}_{+} \:{by}\:\:\: \\ $$$${I}_{\frac{\mathrm{1}}{{n}}} \left({f}\right)\left({x}\right)=\:{a}_{{n}} \int_{\mathrm{0}} ^{{x}} \:\:\frac{{f}\left({t}\right)}{\left({x}−{t}\right)^{\mathrm{1}−\frac{\mathrm{1}}{{n}}} }{dt}\:\:\:\:\:\:\:{and}\:\:{D}_{\frac{\mathrm{1}}{{n}}} \left({f}\right)\:=\:\left({I}_{\frac{\mathrm{1}}{{n}}} \left({f}\right)\right)^{\left(\mathrm{1}\right)} \:\: \\ $$$$\left.\mathrm{1}\right)\:{a}\:\_\:\:{Prove}\:{that}\:{I}_{\frac{\mathrm{1}}{{n}}\:} \left({f}\right)\left({x}\right)=\:{na}_{{n}} {x}^{\frac{\mathrm{1}}{{n}}} \:\int_{\mathrm{0}} ^{\mathrm{1}} \:\:{f}\left({x}\left(\mathrm{1}−{v}^{{n}} \right)\right){dv}\:\:\:{then}\:{find}\:{D}_{\frac{\mathrm{1}}{\mathrm{2}}} \left({t}\right) \\ $$$$\left.\:\:{b}\right)\:\:{Show}\:{that}\:\:\forall\:{f}\in{C}^{\mathrm{1}} \left(\mathbb{R},\mathbb{R}\right)\:\forall\:{x}\in\mathbb{R}_{+\:\:} \:{D}_{\frac{\mathrm{1}}{{n}}} \left({f}\right)\left({x}\right)=\:{I}_{\frac{\mathrm{1}}{{n}}} \left({f}\right)\left({x}\right)\:\:+\:\frac{{f}\left(\mathrm{0}\right)}{\left(\pi{x}\right)^{\mathrm{1}−\frac{\mathrm{1}}{{n}}} } \\ $$$$\left.\mathrm{2}\right)\forall\:{p}\:{integer}\:{and}\:\:{k}\in\left\{\mathrm{0},...,{n}−\mathrm{1}\right\}\:\:{explicit}\:\:{I}_{\frac{\mathrm{1}}{{n}}} \left({t}^{{p}+\frac{{k}}{{n}}} \right)\:{in}\:{term}\:{of}\:\:{I}_{{n},{k}} \left({p}\right) \\ $$$$\left.{b}\right)\:{Prove}\:{that}\:{for}\:{polynomial}\:{function}\:{f}\:\:\:{the}\:{n}−\:{th}\:{composition}\:\:{I}_{\frac{\mathrm{1}}{{n}}} ._{} ....{I}_{\frac{\mathrm{1}}{{n}}} \left({f}\right)\left({x}\right)=\int_{\mathrm{0}} ^{{x}} {f}\left({t}\right){dt}\:\:\:\:\:,\:\: \\ $$$$\left.\:{c}\right)\:{Deduce}\:{that}\:\forall\:\:{f}\:\:{polynomial}\:\:{the}\:{function}\:{g}\:={f}\:−{f}\left(\mathrm{0}\right)\:{verify} \\ $$$${D}_{\frac{\mathrm{1}}{{n}}} ......{D}_{\frac{\mathrm{1}}{{n}}} \left({g}\right)\left({x}\right)\:=\:{g}\left({x}\right) \\ $$$$\left.\mathrm{3}\right)\:{Widen}\:{that}\:{two}\:{formulas}\:{to}\:{all}\:\:{function}\:{that}\:{can}\:{be}\:{developp}\:{into}\:{integer}\:{serie} \\ $$$$\left.\mathrm{4}\right)\:{Try}\:{to}\:{find}\:{the}\:{relation}\:{between}\:\:{D}_{\frac{\mathrm{1}}{{n}}} .{I}_{\frac{\mathrm{1}}{{n}}} \left({f}\right)\:,\:\:{I}_{\frac{\mathrm{1}}{{n}}} .{D}_{\frac{\mathrm{1}}{{n}}} \left({f}\right),\:{and}\:\:{f}\:\: \\ $$$$\left.\mathrm{4}\right)\:{Show}\:\:{that}\:\forall\:{x}\in\mathbb{R}_{+} \:\:{lim}_{{n}−>\infty} \:\:{I}_{\frac{\mathrm{1}}{{n}}} \left({f}\right)\left({x}\right)=\:\int_{\mathrm{0}} ^{{x}} \:{f}\left({t}\right){dt}\:\:\:\:\:\:\: \\ $$$${pour}\:\:{g}={f}−{f}\left(\mathrm{0}\right)\:\:\:{lim}_{{n}−>\infty} \:\:{D}_{\frac{\mathrm{1}}{{n}}} \left({g}\right)\left({x}\right)=\:{g}\left({x}\right) \\ $$$${conclusion} \\ $$$$\:{the}\:{derivative}\:{of}\:{the}\:{function}\:{I}_{\alpha} \:\left({f}\right)\:\:{defined}\:{on}\:\mathbb{R}_{+} \:\:{by}\: \\ $$$${I}_{\alpha} \left({f}\right)\left({x}\right)=\:{a}_{{n}} \int_{\mathrm{0}} ^{{x}} \:{f}\left({t}\right)\left({x}−{t}\right)^{\frac{\mathrm{1}}{{n}}−\mathrm{1}} {dt}\:\:{is}\:\:{called}\:{the}\:\:{derivative}\:{of}\:{order}\:\alpha \\ $$$$ \\ $$

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

$$\mathrm{If}\:{xyz}\:\neq\:\mathrm{0}\:\mathrm{and}\:{x}+{y}+{z}=\mathrm{0} \\ $$$${a}=\mathrm{10}^{{z}} \\ $$$${b}=\mathrm{10}^{{y}} \\ $$$${c}=\mathrm{10}^{{x}} \\ $$$$\mathrm{then} \\ $$$${a}^{\left(\frac{\mathrm{1}}{{y}}+\frac{\mathrm{1}}{{z}}\right)} .\:{b}^{\left(\frac{\mathrm{1}}{{z}}+\frac{\mathrm{1}}{{x}}\right)} .{c}^{\left(\frac{\mathrm{1}}{{x}}+\frac{\mathrm{1}}{{y}}\right)} =... \\ $$$${a}.\:\mathrm{0}.\mathrm{001} \\ $$$${b}.\:\mathrm{0}.\mathrm{01} \\ $$$${c}.\:\mathrm{0}.\mathrm{1} \\ $$$${d}.\:\mathrm{1} \\ $$$${e}.\:\mathrm{10} \\ $$$$ \\ $$

Question Number 65779    Answers: 0   Comments: 1

calculate lim_(x→0) ((sin(x^2 )−xtan(x))/(1−cos(4x)))

$${calculate}\:{lim}_{{x}\rightarrow\mathrm{0}} \:\:\:\:\:\frac{{sin}\left({x}^{\mathrm{2}} \right)−{xtan}\left({x}\right)}{\mathrm{1}−{cos}\left(\mathrm{4}{x}\right)} \\ $$

Question Number 65778    Answers: 0   Comments: 1

find lim_(n→+∞) e^(−n^2 ) (n+1)^(n!)

$${find}\:{lim}_{{n}\rightarrow+\infty} \:{e}^{−{n}^{\mathrm{2}} } \left({n}+\mathrm{1}\right)^{{n}!} \\ $$

Question Number 65777    Answers: 0   Comments: 0

find lim_(n→+∞) e^(−n) ((n+1)!)^n

$${find}\:{lim}_{{n}\rightarrow+\infty} \:\:{e}^{−{n}} \left(\left({n}+\mathrm{1}\right)!\right)^{{n}} \\ $$

Question Number 65776    Answers: 0   Comments: 1

find ∫_(−(π/4)) ^(π/4) ((cosx)/(2+5sinx))dx

$${find}\:\:\int_{−\frac{\pi}{\mathrm{4}}} ^{\frac{\pi}{\mathrm{4}}} \:\:\frac{{cosx}}{\mathrm{2}+\mathrm{5}{sinx}}{dx} \\ $$

Question Number 65775    Answers: 0   Comments: 1

calculate ∫_0 ^(2π) ((tanx)/(2+3cosx))dx

$$\:{calculate}\:\int_{\mathrm{0}} ^{\mathrm{2}\pi} \:\:\:\frac{{tanx}}{\mathrm{2}+\mathrm{3}{cosx}}{dx} \\ $$

Question Number 65774    Answers: 0   Comments: 0

find A_n = ∫_0 ^(2π) ((sin^2 x)/(sin^2 (((nx)/2))))dx (n>0)

$${find}\:{A}_{{n}} =\:\int_{\mathrm{0}} ^{\mathrm{2}\pi} \:\:\:\:\:\frac{{sin}^{\mathrm{2}} {x}}{{sin}^{\mathrm{2}} \left(\frac{{nx}}{\mathrm{2}}\right)}{dx}\:\:\:\:\left({n}>\mathrm{0}\right) \\ $$

Question Number 65773    Answers: 0   Comments: 2

let f(x) =∫_0 ^1 (dt/(1+x(√(1+t^2 )))) with x>0 1)detemine a explicit form of f(x) 2)find also g(x) =∫_0 ^1 ((√(1+t^2 ))/((1+x(√(1+t^2 )))^2 ))dt 3) find the value of integrals ∫_0 ^1 (dt/(1+2(√(1+t^2 )))) and ∫_0 ^1 (dt/((1+2(√(1+t^2 )))^2 ))

$${let}\:{f}\left({x}\right)\:=\int_{\mathrm{0}} ^{\mathrm{1}} \:\:\frac{{dt}}{\mathrm{1}+{x}\sqrt{\mathrm{1}+{t}^{\mathrm{2}} }}\:\:{with}\:{x}>\mathrm{0} \\ $$$$\left.\mathrm{1}\right){detemine}\:{a}\:{explicit}\:{form}\:{of}\:{f}\left({x}\right) \\ $$$$\left.\mathrm{2}\right){find}\:{also}\:\:{g}\left({x}\right)\:=\int_{\mathrm{0}} ^{\mathrm{1}} \:\:\frac{\sqrt{\mathrm{1}+{t}^{\mathrm{2}} }}{\left(\mathrm{1}+{x}\sqrt{\mathrm{1}+{t}^{\mathrm{2}} }\right)^{\mathrm{2}} }{dt} \\ $$$$\left.\mathrm{3}\right)\:{find}\:{the}\:{value}\:{of}\:{integrals}\:\:\int_{\mathrm{0}} ^{\mathrm{1}} \:\frac{{dt}}{\mathrm{1}+\mathrm{2}\sqrt{\mathrm{1}+{t}^{\mathrm{2}} }}\:{and}\: \\ $$$$\int_{\mathrm{0}} ^{\mathrm{1}} \:\:\:\:\:\frac{{dt}}{\left(\mathrm{1}+\mathrm{2}\sqrt{\mathrm{1}+{t}^{\mathrm{2}} }\right)^{\mathrm{2}} } \\ $$

Question Number 65771    Answers: 0   Comments: 0

let X_n =∫_0 ^(π/4) sin^n xdx 1) calculate X_0 ,X_1 ,X_2 ,X_3 2) find X_n interms of n 3)find the value of ∫_0 ^(π/4) sin^8 xdx

$${let}\:{X}_{{n}} =\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{4}}} \:{sin}^{{n}} {xdx} \\ $$$$\left.\mathrm{1}\right)\:{calculate}\:{X}_{\mathrm{0}} \:,{X}_{\mathrm{1}} \:,{X}_{\mathrm{2}} ,{X}_{\mathrm{3}} \\ $$$$\left.\mathrm{2}\right)\:{find}\:{X}_{{n}} {interms}\:{of}\:{n} \\ $$$$\left.\mathrm{3}\right){find}\:{the}\:{value}\:{of}\:\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{4}}} \:{sin}^{\mathrm{8}} {xdx} \\ $$

Question Number 65770    Answers: 0   Comments: 2

let A_n =∫_0 ^(π/2) cos^n xdx 1) calculate A_0 ,A_2 and A_3 2)calculate A_n interms of n 3) find ∫_0 ^(π/2) cos^8 xdx

$${let}\:{A}_{{n}} =\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \:{cos}^{{n}} {xdx} \\ $$$$\left.\mathrm{1}\right)\:{calculate}\:{A}_{\mathrm{0}} ,{A}_{\mathrm{2}} \:{and}\:{A}_{\mathrm{3}} \\ $$$$\left.\mathrm{2}\right){calculate}\:{A}_{{n}} {interms}\:{of}\:{n} \\ $$$$\left.\mathrm{3}\right)\:{find}\:\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \:{cos}^{\mathrm{8}} {xdx}\: \\ $$

Question Number 65769    Answers: 0   Comments: 3

find the value of ∫_0 ^∞ (dx/((x^2 −2xcosθ +1)^2 ))

$${find}\:{the}\:{value}\:{of}\:\int_{\mathrm{0}} ^{\infty} \:\:\:\frac{{dx}}{\left({x}^{\mathrm{2}} −\mathrm{2}{xcos}\theta\:+\mathrm{1}\right)^{\mathrm{2}} } \\ $$

Question Number 65768    Answers: 0   Comments: 1

find the value of ∫_0 ^∞ (dx/(x^2 −2(cosθ)x +1))

$${find}\:{the}\:{value}\:{of}\:\:\int_{\mathrm{0}} ^{\infty} \:\:\:\frac{{dx}}{{x}^{\mathrm{2}} −\mathrm{2}\left({cos}\theta\right){x}\:+\mathrm{1}} \\ $$

Question Number 65767    Answers: 0   Comments: 0

let f(x) =∫_0 ^(+∞) (dt/(t^4 +x^4 )) with x>0 1) determine a explicit form of f(x) 2) find also g(x) =∫_0 ^∞ (dt/((t^4 +x^4 )^2 )) 3)give f^((n)) (x) at form of integral 4) calculate ∫_0 ^∞ (dt/(t^4 +8)) and ∫_0 ^∞ (dt/((t^4 +8)^2 )) 5) calculate A_n =∫_0 ^∞ (dt/((t^4 +x^4 )^n )) with n integr natural

$${let}\:\:{f}\left({x}\right)\:=\int_{\mathrm{0}} ^{+\infty} \:\:\:\frac{{dt}}{{t}^{\mathrm{4}} +{x}^{\mathrm{4}} }\:\:{with}\:{x}>\mathrm{0} \\ $$$$\left.\mathrm{1}\right)\:{determine}\:{a}\:{explicit}\:{form}\:{of}\:{f}\left({x}\right) \\ $$$$\left.\mathrm{2}\right)\:{find}\:{also}\:{g}\left({x}\right)\:=\int_{\mathrm{0}} ^{\infty} \:\:\:\frac{{dt}}{\left({t}^{\mathrm{4}} \:+{x}^{\mathrm{4}} \right)^{\mathrm{2}} } \\ $$$$\left.\mathrm{3}\right){give}\:{f}^{\left({n}\right)} \left({x}\right)\:{at}\:{form}\:{of}\:{integral} \\ $$$$\left.\mathrm{4}\right)\:{calculate}\:\int_{\mathrm{0}} ^{\infty} \:\:\frac{{dt}}{{t}^{\mathrm{4}} \:+\mathrm{8}}\:\:{and}\:\int_{\mathrm{0}} ^{\infty} \:\:\frac{{dt}}{\left({t}^{\mathrm{4}} \:+\mathrm{8}\right)^{\mathrm{2}} } \\ $$$$\left.\mathrm{5}\right)\:{calculate}\:{A}_{{n}} =\int_{\mathrm{0}} ^{\infty} \:\:\:\:\frac{{dt}}{\left({t}^{\mathrm{4}} \:+{x}^{\mathrm{4}} \right)^{{n}} }\:\:{with}\:{n}\:{integr}\:{natural} \\ $$

Question Number 65763    Answers: 0   Comments: 0

Question Number 65753    Answers: 1   Comments: 0

Question Number 65760    Answers: 1   Comments: 2

∫_( 0) ^(π/2) (1/(9 cos x+12 sin x)) dx =

$$\underset{\:\mathrm{0}} {\overset{\pi/\mathrm{2}} {\int}}\:\frac{\mathrm{1}}{\mathrm{9}\:\mathrm{cos}\:{x}+\mathrm{12}\:\mathrm{sin}\:{x}}\:{dx}\:= \\ $$

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