Question and Answers Forum

AllQuestion and Answers: Page 1786

Question Number 30776 Answers: 1 Comments: 1

find ∫_0 ^1 ((xdx)/((1+x^2 )(√(1−x^4 )))) .

$${find}\:\int_{\mathrm{0}} ^{\mathrm{1}} \:\:\:\frac{{xdx}}{\left(\mathrm{1}+{x}^{\mathrm{2}} \right)\sqrt{\mathrm{1}−{x}^{\mathrm{4}} }}\:. \\ $$

Question Number 30775 Answers: 1 Comments: 1

letα ∈]0,π[ calculate ∫_0 ^(π/2) (dx/(2(cosα +chx))) .

$$\left.{let}\alpha\:\in\right]\mathrm{0},\pi\left[\:\:{calculate}\:\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \:\:\:\frac{{dx}}{\mathrm{2}\left({cos}\alpha\:+{chx}\right)}\:.\right. \\ $$

Question Number 30774 Answers: 1 Comments: 1

find f(t)=∫_0 ^1 ln(1+tx^2 )dx with t>0

$${find}\:{f}\left({t}\right)=\int_{\mathrm{0}} ^{\mathrm{1}} \:{ln}\left(\mathrm{1}+{tx}^{\mathrm{2}} \right){dx}\:\:{with}\:{t}>\mathrm{0} \\ $$

Question Number 30773 Answers: 0 Comments: 1

let a>0 calculate ∫_0 ^∞ (dx/((x^2 +a^2 )^2 )) 2) calculate ∫_(−∞) ^(+∞) (x^2 /((x^2 +a^2 )^2 ))dx.

$${let}\:{a}>\mathrm{0}\:{calculate}\:\int_{\mathrm{0}} ^{\infty} \:\:\:\:\:\frac{{dx}}{\left({x}^{\mathrm{2}} \:+{a}^{\mathrm{2}} \right)^{\mathrm{2}} } \\ $$$$\left.\mathrm{2}\right)\:{calculate}\:\:\int_{−\infty} ^{+\infty} \:\:\:\frac{{x}^{\mathrm{2}} }{\left({x}^{\mathrm{2}} \:+{a}^{\mathrm{2}} \right)^{\mathrm{2}} }{dx}. \\ $$

Question Number 30771 Answers: 0 Comments: 0

let I_n = ∫_0 ^(π/4) (dx/(cos^(2n+1) )) (n∈N) 1) find a and b fromR /∀x∈[0,(π/4)] (1/(cosx))=((acosx)/(1−sinx)) +((bcosx)/(1+sinx)) .find I_0 2) verify the relation (1/(cos^(2n+3) x))=(1/(cos^(2n+1) x)) +((sinx sinx)/(cos^(2n+3) )) .find the relation of recurrence between I_n and I_(n+1) .

$${let}\:\:{I}_{{n}} =\:\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{4}}} \:\:\frac{{dx}}{{cos}^{\mathrm{2}{n}+\mathrm{1}} }\:\:\:\:\left({n}\in{N}\right) \\ $$$$\left.\mathrm{1}\right)\:{find}\:{a}\:{and}\:{b}\:{fromR}\:/\forall{x}\in\left[\mathrm{0},\frac{\pi}{\mathrm{4}}\right] \\ $$$$\frac{\mathrm{1}}{{cosx}}=\frac{{acosx}}{\mathrm{1}−{sinx}}\:+\frac{{bcosx}}{\mathrm{1}+{sinx}}\:\:.{find}\:\:{I}_{\mathrm{0}} \\ $$$$\left.\mathrm{2}\right)\:{verify}\:{the}\:{relation} \\ $$$$\frac{\mathrm{1}}{{cos}^{\mathrm{2}{n}+\mathrm{3}} {x}}=\frac{\mathrm{1}}{{cos}^{\mathrm{2}{n}+\mathrm{1}} {x}}\:+\frac{{sinx}\:{sinx}}{{cos}^{\mathrm{2}{n}+\mathrm{3}} }\:.{find}\:{the}\:{relation} \\ $$$${of}\:{recurrence}\:{between}\:{I}_{{n}} \:{and}\:{I}_{{n}+\mathrm{1}} \:\:. \\ $$

Question Number 30769 Answers: 0 Comments: 1

find the value of I= ∫_0 ^1 (dx/((x+1)^2 (√(x^2 +2x +2)))) .

$${find}\:{the}\:{value}\:{of}\:{I}=\:\int_{\mathrm{0}} ^{\mathrm{1}} \:\:\:\:\:\frac{{dx}}{\left({x}+\mathrm{1}\right)^{\mathrm{2}} \sqrt{{x}^{\mathrm{2}} \:+\mathrm{2}{x}\:+\mathrm{2}}}\:. \\ $$

Question Number 30767 Answers: 0 Comments: 1

calculate A_n = ∫_0 ^1 ((1−(√x))/(1−^n (√x)))dx.

$${calculate}\:\:{A}_{{n}} =\:\int_{\mathrm{0}} ^{\mathrm{1}} \:\:\frac{\mathrm{1}−\sqrt{{x}}}{\mathrm{1}−^{{n}} \sqrt{{x}}}{dx}. \\ $$

Question Number 30766 Answers: 0 Comments: 0

find ∫_0 ^1 (x/(√(x^4 +x^2 +1)))dx

$${find}\:\:\:\int_{\mathrm{0}} ^{\mathrm{1}} \:\:\:\:\:\:\frac{{x}}{\sqrt{{x}^{\mathrm{4}} \:+{x}^{\mathrm{2}} \:+\mathrm{1}}}{dx} \\ $$

Question Number 30765 Answers: 0 Comments: 1

find ∫_0 ^1 (dx/(√(x^2 +x+1))) .

$${find}\:\:\int_{\mathrm{0}} ^{\mathrm{1}} \:\:\:\:\:\:\frac{{dx}}{\sqrt{{x}^{\mathrm{2}} \:+{x}+\mathrm{1}}}\:. \\ $$

Question Number 30764 Answers: 0 Comments: 1

let I_n = ∫_0 ^(1/2) (1−2t)^n e^(−t) dt with n integr not 0 1) prove that ∀t∈[0,(1/2)] (1/(√e))(1−2t)^n ≤ (1−2t)^n e^(−t) ≤(1−2t)^n then find lim_(n→∞ ) I_n 2) prove that I_(n+1 ) =1−2(n+1)I_n 3) calculate I_1 ,I_2 , and I_3 .

$${let}\:\:{I}_{{n}} =\:\int_{\mathrm{0}} ^{\frac{\mathrm{1}}{\mathrm{2}}} \:\left(\mathrm{1}−\mathrm{2}{t}\right)^{{n}} \:{e}^{−{t}} {dt}\:\:{with}\:{n}\:{integr}\:{not}\:\mathrm{0} \\ $$$$\left.\mathrm{1}\right)\:{prove}\:{that}\:\forall{t}\in\left[\mathrm{0},\frac{\mathrm{1}}{\mathrm{2}}\right] \\ $$$$\frac{\mathrm{1}}{\sqrt{{e}}}\left(\mathrm{1}−\mathrm{2}{t}\right)^{{n}} \leqslant\:\left(\mathrm{1}−\mathrm{2}{t}\right)^{{n}} \:{e}^{−{t}} \:\leqslant\left(\mathrm{1}−\mathrm{2}{t}\right)^{{n}} \:{then}\:{find}\:{lim}_{{n}\rightarrow\infty\:} {I}_{{n}} \\ $$$$\left.\mathrm{2}\right)\:{prove}\:{that}\:{I}_{{n}+\mathrm{1}\:} =\mathrm{1}−\mathrm{2}\left({n}+\mathrm{1}\right){I}_{{n}} \\ $$$$\left.\mathrm{3}\right)\:{calculate}\:{I}_{\mathrm{1}} ,{I}_{\mathrm{2}} ,\:{and}\:{I}_{\mathrm{3}} . \\ $$

Question Number 30761 Answers: 0 Comments: 1

find ∫_0 ^∞ x^(2n+1) e^(−x^2 ) dx with n from N.

$${find}\:\int_{\mathrm{0}} ^{\infty} \:\:{x}^{\mathrm{2}{n}+\mathrm{1}} \:{e}^{−{x}^{\mathrm{2}} } {dx}\:\:\:{with}\:{n}\:{from}\:{N}. \\ $$

Question Number 30760 Answers: 0 Comments: 1

find I_n = ∫_0 ^1 (lnx)^n dx with n fromN

$${find}\:\:{I}_{{n}} =\:\int_{\mathrm{0}} ^{\mathrm{1}} \:\left({lnx}\right)^{{n}} \:{dx}\:\:{with}\:{n}\:{fromN} \\ $$

Question Number 30759 Answers: 0 Comments: 1

find lim_(n→∞) (((n!)/n^n ))^(1/n) .

$${find}\:{lim}_{{n}\rightarrow\infty} \:\:\:\:\left(\frac{{n}!}{{n}^{{n}} }\right)^{\frac{\mathrm{1}}{{n}}} \:. \\ $$

Question Number 30758 Answers: 0 Comments: 1

find lim_(n→∞) ((1/(n+1)) +(1/(n+2)) +....+(1/(n+p))) pfixed fromN^★

$${find}\:{lim}_{{n}\rightarrow\infty} \left(\frac{\mathrm{1}}{{n}+\mathrm{1}}\:+\frac{\mathrm{1}}{{n}+\mathrm{2}}\:+....+\frac{\mathrm{1}}{{n}+{p}}\right)\:{pfixed}\:{fromN}^{\bigstar} \\ $$$$ \\ $$

Question Number 30757 Answers: 0 Comments: 1

find lim_(n→∞) ((1/n) +(1/(√(n^2 −1))) +.... +(1/(√(n^2 −(n−1)^2 ))) )

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

Question Number 30755 Answers: 0 Comments: 1

find lim_(n→∞) ^n (√((1+(1/n))(1+(2/n))...(1+(n/n)) ))

$${find}\:\:\:{lim}_{{n}\rightarrow\infty} \:\:^{{n}} \sqrt{\left(\mathrm{1}+\frac{\mathrm{1}}{{n}}\right)\left(\mathrm{1}+\frac{\mathrm{2}}{{n}}\right)...\left(\mathrm{1}+\frac{{n}}{{n}}\right)\:} \\ $$

Question Number 30754 Answers: 0 Comments: 0

prove that e^x =Σ_(k=0) ^n (x^k /(k!)) +(x^(n+1) /(n!)) ∫_0 ^ (1−t)^n e^(tx) dt 2) prove that e^x = Σ_(k=0) ^(∞ ) (x^k /(k!)) .

$${prove}\:{that}\:{e}^{{x}} =\sum_{{k}=\mathrm{0}} ^{{n}} \:\frac{{x}^{{k}} }{{k}!}\:+\frac{{x}^{{n}+\mathrm{1}} }{{n}!}\:\int_{\mathrm{0}} ^{} \left(\mathrm{1}−{t}\right)^{{n}} \:{e}^{{tx}} {dt} \\ $$$$\left.\mathrm{2}\right)\:{prove}\:{that}\:\:{e}^{{x}} =\:\sum_{{k}=\mathrm{0}} ^{\infty\:} \:\:\frac{{x}^{{k}} }{{k}!}\:. \\ $$

Question Number 30753 Answers: 0 Comments: 0

let f_n (x)=(1/x^(n+1) ) (e^x −Σ_(p=0) ^(n ) (x^p /(p!))) 1) prove that f^((n)) (x)=((Q_n (x) e^x −P_n (x))/x^(2n+1) ) find the polynomial P_n and Q_n . 2) prove that e^x −((P_n (x))/(Q_n (x)))=o(x^(2n+1) )

$${let}\:{f}_{{n}} \left({x}\right)=\frac{\mathrm{1}}{{x}^{{n}+\mathrm{1}} }\:\left({e}^{{x}} \:\:−\sum_{{p}=\mathrm{0}} ^{{n}\:} \:\:\:\frac{{x}^{{p}} }{{p}!}\right) \\ $$$$\left.\mathrm{1}\right)\:{prove}\:{that}\:{f}^{\left({n}\right)} \left({x}\right)=\frac{{Q}_{{n}} \left({x}\right)\:{e}^{{x}} \:−{P}_{{n}} \left({x}\right)}{{x}^{\mathrm{2}{n}+\mathrm{1}} }\:{find}\:{the} \\ $$$${polynomial}\:{P}_{{n}} \:{and}\:{Q}_{{n}} . \\ $$$$\left.\mathrm{2}\right)\:{prove}\:{that}\:{e}^{{x}} \:−\frac{{P}_{{n}} \left({x}\right)}{{Q}_{{n}} \left({x}\right)}={o}\left({x}^{\mathrm{2}{n}+\mathrm{1}} \:\:\right) \\ $$

Question Number 30752 Answers: 0 Comments: 1

let f(x)= (1/(1+x^2 )) calculate f^((n)) (x).

$${let}\:{f}\left({x}\right)=\:\frac{\mathrm{1}}{\mathrm{1}+{x}^{\mathrm{2}} }\:\:{calculate}\:{f}^{\left({n}\right)} \left({x}\right). \\ $$

Question Number 30751 Answers: 0 Comments: 0

study and give the graph for f(x)=(x^2 /(x−1)) e^(1/x) .

$${study}\:{and}\:{give}\:{the}\:{graph}\:{for}\: \\ $$$${f}\left({x}\right)=\frac{{x}^{\mathrm{2}} }{{x}−\mathrm{1}}\:{e}^{\frac{\mathrm{1}}{{x}}} \:\:. \\ $$

Question Number 30750 Answers: 0 Comments: 0

f function C^∞ /f^′ =1+f^2 let take a_k =((f^((k)) (0))/(k!)) prove that a_(n+1) = (1/(n+1)) Σ_(k=0) ^n a_k a_(n−k)

$${f}\:{function}\:{C}^{\infty} \:/{f}^{'} =\mathrm{1}+{f}^{\mathrm{2}} \:\:{let}\:{take}\:{a}_{{k}} =\frac{{f}^{\left({k}\right)} \left(\mathrm{0}\right)}{{k}!}\: \\ $$$${prove}\:{that}\:{a}_{{n}+\mathrm{1}} =\:\frac{\mathrm{1}}{{n}+\mathrm{1}}\:\sum_{{k}=\mathrm{0}} ^{{n}} \:{a}_{{k}} \:\:{a}_{{n}−{k}} \\ $$

Question Number 30749 Answers: 0 Comments: 0

let f(x)=arcsinx with x∈[0,1] 1) prove that (1−x^2 )f^(′′) (x) −xf^′ (x)=0 2)prove that (1−x^2 )f^((n+2)) (x)=(2n+1)x f^((n+1)) (x) +n^2 f^((n)) (x) 3) prove that f^((n)) (x) ≥0 ∀n .

$${let}\:{f}\left({x}\right)={arcsinx}\:{with}\:{x}\in\left[\mathrm{0},\mathrm{1}\right] \\ $$$$\left.\mathrm{1}\right)\:{prove}\:{that}\:\left(\mathrm{1}−{x}^{\mathrm{2}} \right){f}^{''} \left({x}\right)\:−{xf}^{'} \left({x}\right)=\mathrm{0} \\ $$$$\left.\mathrm{2}\right){prove}\:{that}\:\left(\mathrm{1}−{x}^{\mathrm{2}} \right){f}^{\left({n}+\mathrm{2}\right)} \left({x}\right)=\left(\mathrm{2}{n}+\mathrm{1}\right){x}\:{f}^{\left({n}+\mathrm{1}\right)} \left({x}\right)\:+{n}^{\mathrm{2}} {f}^{\left({n}\right)} \left({x}\right) \\ $$$$\left.\mathrm{3}\right)\:{prove}\:{that}\:\:{f}^{\left({n}\right)} \left({x}\right)\:\geqslant\mathrm{0}\:\forall{n}\:. \\ $$

Question Number 30748 Answers: 0 Comments: 0

let a>0 and b>0 find lim_(x→0^+ ) ( ((a^x +b^x )/2))^(1/x) .

$${let}\:{a}>\mathrm{0}\:{and}\:{b}>\mathrm{0}\:{find}\:{lim}_{{x}\rightarrow\mathrm{0}^{+} } \:\:\left(\:\:\frac{{a}^{{x}} \:+{b}^{{x}} }{\mathrm{2}}\right)^{\frac{\mathrm{1}}{{x}}} \:\:. \\ $$

Question Number 30747 Answers: 0 Comments: 0

let f(x)=(√(1+x^2 )) 1)find a d.e.wich verify f(x) 2) prove that ∀x∈R ,∀n∈N (1+x^2 )f^((n+2)) (x)+(2n+1)x f^((n+1)) (x) +(n^2 −1)f^((n)) (x)=0 3) prove that f^((2n+1)) (0)=0 ∀n∈ N

$${let}\:{f}\left({x}\right)=\sqrt{\mathrm{1}+{x}^{\mathrm{2}} } \\ $$$$\left.\mathrm{1}\right){find}\:{a}\:{d}.{e}.{wich}\:{verify}\:{f}\left({x}\right) \\ $$$$\left.\mathrm{2}\right)\:{prove}\:{that}\:\forall{x}\in{R}\:,\forall{n}\in{N} \\ $$$$\left(\mathrm{1}+{x}^{\mathrm{2}} \right){f}^{\left({n}+\mathrm{2}\right)} \left({x}\right)+\left(\mathrm{2}{n}+\mathrm{1}\right){x}\:{f}^{\left({n}+\mathrm{1}\right)} \left({x}\right)\:+\left({n}^{\mathrm{2}} −\mathrm{1}\right){f}^{\left({n}\right)} \left({x}\right)=\mathrm{0} \\ $$$$\left.\mathrm{3}\right)\:{prove}\:{that}\:\:{f}^{\left(\mathrm{2}{n}+\mathrm{1}\right)} \left(\mathrm{0}\right)=\mathrm{0}\:\forall{n}\in\:{N} \\ $$

Question Number 30745 Answers: 0 Comments: 0

let U_n ={z∈C/z^n =1} simlify A_n = Σ_(α∈U_n ) (x+α)^n and B_n =Σ_(α∈ U_n ) (x−α)^n .

$${let}\:\:{U}_{{n}} =\left\{{z}\in{C}/{z}^{{n}} =\mathrm{1}\right\}\:\:{simlify} \\ $$$${A}_{{n}} =\:\sum_{\alpha\in{U}_{{n}} } \:\left({x}+\alpha\right)^{{n}} \:{and}\:{B}_{{n}} =\sum_{\alpha\in\:{U}_{{n}} } \:\:\left({x}−\alpha\right)^{{n}} . \\ $$

Question Number 30744 Answers: 0 Comments: 0

let p(x)= (x−1)^n −x^n +1 with n integr find n in ordre that p(x) have a double root.

$${let}\:{p}\left({x}\right)=\:\left({x}−\mathrm{1}\right)^{{n}} \:−{x}^{{n}} \:+\mathrm{1}\:\:{with}\:{n}\:{integr}\:{find}\:{n} \\ $$$${in}\:{ordre}\:{that}\:{p}\left({x}\right)\:{have}\:{a}\:{double}\:{root}. \\ $$

Pg 1781 Pg 1782 Pg 1783 Pg 1784 Pg 1785 Pg 1786 Pg 1787 Pg 1788 Pg 1789 Pg 1790

Contact: info@tinkutara.com