Question and Answers Forum

All Questions   Topic List

AllQuestion and Answers: Page 1674

Question Number 38203    Answers: 0   Comments: 0

let x≠(π/2)+kπ,k∈Z.prove that (1/(2cosx)) =Σ_(n=0) ^∞ (−1)^n (cos(2n+1)x)

$${let}\:{x}\neq\frac{\pi}{\mathrm{2}}+{k}\pi,{k}\in{Z}.{prove}\:{that} \\ $$$$\frac{\mathrm{1}}{\mathrm{2}{cosx}}\:=\sum_{{n}=\mathrm{0}} ^{\infty} \left(−\mathrm{1}\right)^{{n}} \left({cos}\left(\mathrm{2}{n}+\mathrm{1}\right){x}\right) \\ $$

Question Number 38202    Answers: 0   Comments: 1

find the value of ∫_0 ^1 e^(−x) (√(1−e^(−2x) ))dx

$${find}\:{the}\:{value}\:{of}\:\:\int_{\mathrm{0}} ^{\mathrm{1}} \:{e}^{−{x}} \sqrt{\mathrm{1}−{e}^{−\mathrm{2}{x}} }{dx} \\ $$

Question Number 38201    Answers: 0   Comments: 0

calculate I = ∫_0 ^∞ xe^(−x^2 ) (√(1−e^(−2x^2 ) ))dx

$${calculate}\:{I}\:=\:\int_{\mathrm{0}} ^{\infty} \:\:\:{xe}^{−{x}^{\mathrm{2}} } \sqrt{\mathrm{1}−{e}^{−\mathrm{2}{x}^{\mathrm{2}} } }{dx} \\ $$

Question Number 38199    Answers: 1   Comments: 0

calculate ∫_0 ^1 ((√x)/(1+x^2 ))dx .

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

Question Number 38198    Answers: 0   Comments: 5

we give ∫_0 ^∞ e^(−x) ln(x)dx=−γ 1) calculate f(a)= ∫_0 ^∞ e^(−ax) ln(x)dx with a>0 2) let u_n = ∫_0 ^∞ e^(−nx) ln((x/n))dx find lim_(n→+∞) u_n

$${we}\:{give}\:\int_{\mathrm{0}} ^{\infty} \:\:{e}^{−{x}} {ln}\left({x}\right){dx}=−\gamma \\ $$$$\left.\mathrm{1}\right)\:{calculate}\:\:{f}\left({a}\right)=\:\int_{\mathrm{0}} ^{\infty} \:\:{e}^{−{ax}} {ln}\left({x}\right){dx}\:\:{with}\:{a}>\mathrm{0} \\ $$$$\left.\mathrm{2}\right)\:{let}\:{u}_{{n}} =\:\int_{\mathrm{0}} ^{\infty} \:\:{e}^{−{nx}} {ln}\left(\frac{{x}}{{n}}\right){dx}\:\:{find}\:{lim}_{{n}\rightarrow+\infty} \:{u}_{{n}} \\ $$

Question Number 38197    Answers: 0   Comments: 1

find a simple form of L(e^(−(√x)) ) L is laplace transform

$${find}\:{a}\:{simple}\:{form}\:{of}\:{L}\left({e}^{−\sqrt{{x}}} \right)\:\:{L}\:{is}\:{laplace}\:{transform} \\ $$

Question Number 38195    Answers: 0   Comments: 1

let x≥1 and δ(x)=Σ_(n=1) ^∞ (((−1)^n )/n^x ) 1) calculate δ(x) interms of ξ(x) if x>1 2)find δ(1) 3) find the value of Σ_(n=1) ^∞ (1/((2n+1)^2 )) 4) calculate δ(3) interms of ξ(3).

$${let}\:{x}\geqslant\mathrm{1}\:{and}\:\delta\left({x}\right)=\sum_{{n}=\mathrm{1}} ^{\infty} \:\:\frac{\left(−\mathrm{1}\right)^{{n}} }{{n}^{{x}} } \\ $$$$\left.\mathrm{1}\right)\:{calculate}\:\delta\left({x}\right)\:{interms}\:{of}\:\xi\left({x}\right)\:{if}\:{x}>\mathrm{1} \\ $$$$\left.\mathrm{2}\right){find}\:\:\delta\left(\mathrm{1}\right) \\ $$$$\left.\mathrm{3}\right)\:{find}\:{the}\:{value}\:{of}\:\:\sum_{{n}=\mathrm{1}} ^{\infty} \:\frac{\mathrm{1}}{\left(\mathrm{2}{n}+\mathrm{1}\right)^{\mathrm{2}} } \\ $$$$\left.\mathrm{4}\right)\:{calculate}\:\delta\left(\mathrm{3}\right)\:{interms}\:{of}\:\xi\left(\mathrm{3}\right). \\ $$

Question Number 38190    Answers: 1   Comments: 2

Question Number 38181    Answers: 2   Comments: 3

If y= x^((lnx)^(ln(lnx)) ) then (dy/dx) = ?

$$\mathrm{If}\:\mathrm{y}=\:\:{x}^{\left({lnx}\right)^{{ln}\left({lnx}\right)} } \:{then}\:\frac{{dy}}{{dx}}\:=\:? \\ $$

Question Number 38177    Answers: 0   Comments: 0

Question Number 38155    Answers: 0   Comments: 1

Propanol ,(C_3 H_7 OH) is an alcohol. a) State the functional group in propanol that makes it alcohol. b) Ethanol can be converted to a number of compounds as shown below. Ethene→ Ethane ↑ Ethylethanoate ⇆ Ethanol → Ethanoic acid ↓ soduimethanoate. when Soduim hydroxide solution reacts with Ethylethanoate ,Sodium Ethanoate and other products are obtained Give the name of the other products

$${Propanol}\:,\left({C}_{\mathrm{3}} {H}_{\mathrm{7}} {OH}\right)\:{is}\:{an}\:{alcohol}. \\ $$$$\left.{a}\right)\:{State}\:{the}\:{functional}\:{group}\:{in}\: \\ $$$${propanol}\:{that}\:{makes}\:{it}\:{alcohol}. \\ $$$$\left.{b}\right)\:{Ethanol}\:{can}\:{be}\:{converted}\:{to}\:{a}\:{number} \\ $$$${of}\:{compounds}\:{as}\:{shown}\:{below}. \\ $$$$ \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:{Ethene}\rightarrow\:{Ethane} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\uparrow\: \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:{Ethylethanoate}\:\:\:\:\:\:\leftrightarrows\:\:\:{Ethanol}\:\rightarrow\:{Ethanoic}\:{acid} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\downarrow \\ $$$$\:\:\:\:\:\:\:\:\:\:{soduimethanoate}. \\ $$$${when}\:{Soduim}\:{hydroxide}\:{solution} \\ $$$${reacts}\:{with}\:{Ethylethanoate}\:,{Sodium} \\ $$$${Ethanoate}\:{and}\:{other}\:{products}\:{are}\:{obtained} \\ $$$${Give}\:{the}\:{name}\:{of}\:{the}\:{other}\:{products} \\ $$$$ \\ $$

Question Number 38154    Answers: 1   Comments: 0

Given that a,b,c are 3 consecutive term of a Geometric sequence f(n) , show that log a,logb,logc are the first 3 terms of an Arithmetic SequenceP(n).

$${Given}\:{that}\: \\ $$$$\:\:{a},{b},{c}\:{are}\:\mathrm{3}\:{consecutive}\:{term}\:{of}\: \\ $$$${a}\:{Geometric}\:{sequence}\:{f}\left({n}\right)\:,\:{show} \\ $$$${that}\:{log}\:{a},{logb},{logc}\:{are}\:{the}\:{first}\: \\ $$$$\mathrm{3}\:{terms}\:{of}\:{an}\:{Arithmetic}\:{SequenceP}\left({n}\right). \\ $$

Question Number 38151    Answers: 3   Comments: 8

a > b > 0 a^2 cos θ−b^2 sin θ=(a^2 −b^2 )sin θcos θ Find θ in terms of a, b. θ ∈ (0,(π/2)) .

$${a}\:>\:{b}\:>\:\mathrm{0} \\ $$$${a}^{\mathrm{2}} \mathrm{cos}\:\theta−{b}^{\mathrm{2}} \mathrm{sin}\:\theta=\left({a}^{\mathrm{2}} −{b}^{\mathrm{2}} \right)\mathrm{sin}\:\theta\mathrm{cos}\:\theta \\ $$$${Find}\:\theta\:{in}\:{terms}\:{of}\:{a},\:{b}. \\ $$$$\:\:\theta\:\in\:\left(\mathrm{0},\frac{\pi}{\mathrm{2}}\right)\:. \\ $$

Question Number 38130    Answers: 4   Comments: 0

1. ∫tan^3 (2x)sec^5 (2x) dx 2. ∫_0 ^(π/3) tan^5 (x)sec^6 (x) dx 3. ∫tan^6 (ay) dy

$$\mathrm{1}.\:\int\mathrm{tan}^{\mathrm{3}} \left(\mathrm{2}{x}\right)\mathrm{sec}^{\mathrm{5}} \left(\mathrm{2}{x}\right)\:{dx}\: \\ $$$$\mathrm{2}.\:\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{3}}} \mathrm{tan}^{\mathrm{5}} \left({x}\right)\mathrm{sec}^{\mathrm{6}} \left({x}\right)\:{dx}\: \\ $$$$\mathrm{3}.\:\int\mathrm{tan}^{\mathrm{6}} \left({ay}\right)\:{dy}\: \\ $$

Question Number 38127    Answers: 0   Comments: 1

calculate ∫_0 ^∞ e^(−x) (√(1+e^(−2x) ))dx

$${calculate}\:\int_{\mathrm{0}} ^{\infty} \:\:{e}^{−{x}} \sqrt{\mathrm{1}+{e}^{−\mathrm{2}{x}} }{dx} \\ $$

Question Number 38126    Answers: 1   Comments: 2

calculate ∫_0 ^∞ e^(−2x) (√(1+e^(−4x) ))dx .

$${calculate}\:\:\:\int_{\mathrm{0}} ^{\infty} \:\:{e}^{−\mathrm{2}{x}} \sqrt{\mathrm{1}+{e}^{−\mathrm{4}{x}} }{dx}\:. \\ $$

Question Number 38125    Answers: 1   Comments: 1

let α>0 find ∫_0 ^∞ (e^(−αx) /(√(1+e^(−2αx) )))dx .

$${let}\:\alpha>\mathrm{0}\:{find} \\ $$$$\int_{\mathrm{0}} ^{\infty} \:\:\:\:\:\frac{{e}^{−\alpha{x}} }{\sqrt{\mathrm{1}+{e}^{−\mathrm{2}\alpha{x}} }}{dx}\:. \\ $$

Question Number 38124    Answers: 1   Comments: 0

prove that ∫ (dx/(√(1+x^2 ))) =ln(x+(√(1+x^2 ))) +c 2) find ∫ (dx/(√(a+x^2 ))) with a>0

$${prove}\:{that}\:\:\int\:\:\:\:\:\:\frac{{dx}}{\sqrt{\mathrm{1}+{x}^{\mathrm{2}} }}\:={ln}\left({x}+\sqrt{\mathrm{1}+{x}^{\mathrm{2}} }\right)\:+{c} \\ $$$$\left.\mathrm{2}\right)\:{find}\:\int\:\:\:\frac{{dx}}{\sqrt{{a}+{x}^{\mathrm{2}} }}\:{with}\:{a}>\mathrm{0} \\ $$

Question Number 38123    Answers: 2   Comments: 1

f is a function positive and C^1 1) find ∫ (f^′ /(2(√f)(√(1+f))))dx 2)let A_n = ∫_0 ^1 (x^(n/2) /(x(√(1+x^n )))) calculate A_n and lim_(n→+∞) A_n

$${f}\:{is}\:{a}\:{function}\:{positive}\:\:{and}\:{C}^{\mathrm{1}} \:\: \\ $$$$\left.\mathrm{1}\right)\:{find}\:\int\:\:\:\:\frac{{f}^{'} }{\mathrm{2}\sqrt{{f}}\sqrt{\mathrm{1}+{f}}}{dx} \\ $$$$\left.\mathrm{2}\right){let}\:\:{A}_{{n}} =\:\int_{\mathrm{0}} ^{\mathrm{1}} \:\:\:\:\frac{{x}^{\frac{{n}}{\mathrm{2}}} }{{x}\sqrt{\mathrm{1}+{x}^{{n}} }} \\ $$$${calculate}\:{A}_{{n}} \:\:{and}\:{lim}_{{n}\rightarrow+\infty} \:{A}_{{n}} \\ $$

Question Number 38122    Answers: 1   Comments: 0

calculate ∫_0 ^π ((sinx)/(√(1+cos^2 x)))dx

$${calculate}\:\:\int_{\mathrm{0}} ^{\pi} \:\:\:\:\frac{{sinx}}{\sqrt{\mathrm{1}+{cos}^{\mathrm{2}} {x}}}{dx} \\ $$

Question Number 38121    Answers: 0   Comments: 5

let x>0 find F(x) = ∫_(−∞) ^(+∞) ((arctan(xt^2 ))/(1+t^2 ))dt

$${let}\:{x}>\mathrm{0}\:{find}\:{F}\left({x}\right)\:=\:\int_{−\infty} ^{+\infty} \:\:\:\:\frac{{arctan}\left({xt}^{\mathrm{2}} \right)}{\mathrm{1}+{t}^{\mathrm{2}} }{dt}\: \\ $$

Question Number 38120    Answers: 0   Comments: 2

let n from N and find the value of A_n = ∫_1 ^(+∞) (dt/(t^n (√(t−1))))

$${let}\:\:{n}\:{from}\:{N}\:{and} \\ $$$${find}\:{the}\:{value}\:{of}\:\:{A}_{{n}} =\:\int_{\mathrm{1}} ^{+\infty} \:\:\:\frac{{dt}}{{t}^{{n}} \sqrt{{t}−\mathrm{1}}} \\ $$

Question Number 38119    Answers: 0   Comments: 1

calculate ∫_1 ^(+∞) (dx/(x^4 (√(x−1))))

$${calculate}\:\:\int_{\mathrm{1}} ^{+\infty} \:\:\:\:\:\frac{{dx}}{{x}^{\mathrm{4}} \sqrt{{x}−\mathrm{1}}} \\ $$

Question Number 38118    Answers: 0   Comments: 1

prove that ∫_0 ^1 (1/(1+(t^a /2)))dt =Σ_(n=0) ^∞ (((−1)^n )/(2^n (na+1))) 2) find the value of Σ_(n=0) ^∞ (((−1)^n )/(2^n (3n+1)))

$${prove}\:{that}\:\:\int_{\mathrm{0}} ^{\mathrm{1}} \:\:\:\:\:\:\frac{\mathrm{1}}{\mathrm{1}+\frac{{t}^{{a}} }{\mathrm{2}}}{dt}\:=\sum_{{n}=\mathrm{0}} ^{\infty} \:\:\:\frac{\left(−\mathrm{1}\right)^{{n}} }{\mathrm{2}^{{n}} \left({na}+\mathrm{1}\right)} \\ $$$$\left.\mathrm{2}\right)\:{find}\:{the}\:{value}\:{of}\:\sum_{{n}=\mathrm{0}} ^{\infty} \:\:\:\:\frac{\left(−\mathrm{1}\right)^{{n}} }{\mathrm{2}^{{n}} \left(\mathrm{3}{n}+\mathrm{1}\right)} \\ $$

Question Number 38117    Answers: 0   Comments: 0

let x and y from R prove that ∣cos(x+iy)∣=cos^2 x +sh^2 y ∣sin(x+iy)∣^2 =sin^2 x +sh^2 y

$${let}\:{x}\:{and}\:{y}\:{from}\:{R}\:{prove}\:{that} \\ $$$$\mid{cos}\left({x}+{iy}\right)\mid={cos}^{\mathrm{2}} {x}\:+{sh}^{\mathrm{2}} {y} \\ $$$$\mid{sin}\left({x}+{iy}\right)\mid^{\mathrm{2}} ={sin}^{\mathrm{2}} {x}\:+{sh}^{\mathrm{2}} {y} \\ $$

Question Number 38116    Answers: 0   Comments: 1

find I = ∫_0 ^∞ ((cos(λx))/(ch(2x)))dx

$${find}\:\:\:\:{I}\:=\:\int_{\mathrm{0}} ^{\infty} \:\:\:\frac{{cos}\left(\lambda{x}\right)}{{ch}\left(\mathrm{2}{x}\right)}{dx}\: \\ $$

  Pg 1669      Pg 1670      Pg 1671      Pg 1672      Pg 1673      Pg 1674      Pg 1675      Pg 1676      Pg 1677      Pg 1678   

Terms of Service

Privacy Policy

Contact: info@tinkutara.com