Question and Answers Forum

Relation and FunctionsQuestion and Answers: Page 10

Question Number 144499 Answers: 1 Comments: 0

find ∫_0 ^∞ ((arctan(x^n ))/x^n )dx (n≥2) natural

$$\mathrm{find}\:\int_{\mathrm{0}} ^{\infty} \:\frac{\mathrm{arctan}\left(\mathrm{x}^{\mathrm{n}} \right)}{\mathrm{x}^{\mathrm{n}} }\mathrm{dx}\:\:\:\left(\mathrm{n}\geqslant\mathrm{2}\right)\:\mathrm{natural} \\ $$

Question Number 144464 Answers: 1 Comments: 0

f(x)=(2/((1+sinx)^2 )) developp f at fourier serie

$$\mathrm{f}\left(\mathrm{x}\right)=\frac{\mathrm{2}}{\left(\mathrm{1}+\mathrm{sinx}\right)^{\mathrm{2}} } \\ $$$$\mathrm{developp}\:\mathrm{f}\:\mathrm{at}\:\mathrm{fourier}\:\mathrm{serie} \\ $$

Question Number 144241 Answers: 2 Comments: 0

calculate ∫_0 ^(4π) (dx/((2+cosx)^2 ))

$$\mathrm{calculate}\:\int_{\mathrm{0}} ^{\mathrm{4}\pi} \:\frac{\mathrm{dx}}{\left(\mathrm{2}+\mathrm{cosx}\right)^{\mathrm{2}} } \\ $$

Question Number 144222 Answers: 0 Comments: 0

find ∫_0 ^∞ ((log^2 x)/((x^2 −x+1)^2 ))dx

$${find}\:\int_{\mathrm{0}} ^{\infty} \:\frac{{log}^{\mathrm{2}} {x}}{\left({x}^{\mathrm{2}} −{x}+\mathrm{1}\right)^{\mathrm{2}} }{dx} \\ $$

Question Number 144221 Answers: 0 Comments: 0

find ∫_0 ^∞ e^(−3x) log^2 (1+e^(2x) )dx

$${find}\:\int_{\mathrm{0}} ^{\infty} {e}^{−\mathrm{3}{x}} {log}^{\mathrm{2}} \left(\mathrm{1}+{e}^{\mathrm{2}{x}} \right){dx} \\ $$

Question Number 144220 Answers: 1 Comments: 0

find A_n =∫_0 ^1 arctan(x^n )dx n ∈N

$${find}\:{A}_{{n}} =\int_{\mathrm{0}} ^{\mathrm{1}} \:{arctan}\left({x}^{{n}} \right){dx} \\ $$$${n}\:\in{N} \\ $$

Question Number 144219 Answers: 1 Comments: 0

let f(x)=(1/((2+cosx)^2 )) developp f at fourier serie

$${let}\:{f}\left({x}\right)=\frac{\mathrm{1}}{\left(\mathrm{2}+{cosx}\right)^{\mathrm{2}} } \\ $$$${developp}\:{f}\:{at}\:{fourier}\:{serie} \\ $$

Question Number 144218 Answers: 1 Comments: 0

U_n =Σ_(k=0) ^n (1/( (√(2k+1)))) find a eqivalent of U_n (n→∞)

$${U}_{{n}} =\sum_{{k}=\mathrm{0}} ^{{n}} \frac{\mathrm{1}}{\:\sqrt{\mathrm{2}{k}+\mathrm{1}}} \\ $$$${find}\:{a}\:{eqivalent}\:{of}\:{U}_{{n}} \left({n}\rightarrow\infty\right) \\ $$

Question Number 144217 Answers: 0 Comments: 0

find ∫_0 ^∞ (x^2 /((x+2)^5 (3x+1)^4 ))dx

$${find}\:\int_{\mathrm{0}} ^{\infty} \:\:\:\:\frac{{x}^{\mathrm{2}} }{\left({x}+\mathrm{2}\right)^{\mathrm{5}} \left(\mathrm{3}{x}+\mathrm{1}\right)^{\mathrm{4}} }{dx} \\ $$

Question Number 144216 Answers: 1 Comments: 0

find ∫ (dx/( (√(x^2 +x+2))+(√(x^2 −x+2))))

$${find}\:\int\:\:\:\frac{{dx}}{\:\sqrt{{x}^{\mathrm{2}} +{x}+\mathrm{2}}+\sqrt{{x}^{\mathrm{2}} −{x}+\mathrm{2}}} \\ $$

Question Number 144215 Answers: 0 Comments: 0

find ∫_0 ^∞ e^(−3x) (√(x^2 +x+1))dx

$${find}\:\int_{\mathrm{0}} ^{\infty} \:{e}^{−\mathrm{3}{x}} \sqrt{{x}^{\mathrm{2}} +{x}+\mathrm{1}}{dx} \\ $$

Question Number 144214 Answers: 1 Comments: 0

calculate ∫_0 ^(4π) ((sinx)/((3+cosx)^2 ))dx

$${calculate}\:\int_{\mathrm{0}} ^{\mathrm{4}\pi} \:\:\frac{{sinx}}{\left(\mathrm{3}+{cosx}\right)^{\mathrm{2}} }{dx} \\ $$

Question Number 144213 Answers: 1 Comments: 0

find ∫ (dx/(1+cosx+cos(2x)))

$${find}\:\int\:\:\frac{{dx}}{\mathrm{1}+{cosx}+{cos}\left(\mathrm{2}{x}\right)} \\ $$

Question Number 144212 Answers: 0 Comments: 0

calculate Σ_(n=0) ^∞ (1/(n^3 +1))

$${calculate}\:\sum_{{n}=\mathrm{0}} ^{\infty} \:\frac{\mathrm{1}}{{n}^{\mathrm{3}} +\mathrm{1}} \\ $$

Question Number 144211 Answers: 0 Comments: 0

find Σ_(n=0) ^∞ (1/(n^4 +1))

$${find}\:\sum_{{n}=\mathrm{0}} ^{\infty} \:\frac{\mathrm{1}}{{n}^{\mathrm{4}} +\mathrm{1}} \\ $$

Question Number 144180 Answers: 2 Comments: 0

Prove that ∫^( +∞) _0 ((sh(𝛂t))/(sh(t)))dt = (𝛑/2)tan(((𝛑𝛂)/2))

$$\mathrm{Prove}\:\mathrm{that} \\ $$$$\underset{\mathrm{0}} {\int}^{\:+\infty} \:\frac{\boldsymbol{\mathrm{sh}}\left(\boldsymbol{\alpha\mathrm{t}}\right)}{\boldsymbol{\mathrm{sh}}\left(\boldsymbol{\mathrm{t}}\right)}\boldsymbol{{dt}}\:=\:\frac{\boldsymbol{\pi}}{\mathrm{2}}\boldsymbol{{tan}}\left(\frac{\boldsymbol{\pi\alpha}}{\mathrm{2}}\right) \\ $$

Question Number 144109 Answers: 3 Comments: 1

hi, everybody ! 1. calculate : I =∫_(𝛑/6) ^( (𝛑/3)) ln(tan x)dx. 2. calculate : lim_(x → e) ((x(√(1−ln x)))/(x−e)) .

$$\boldsymbol{\mathrm{hi}},\:\boldsymbol{\mathrm{everybody}}\:! \\ $$$$\mathrm{1}.\:\boldsymbol{\mathrm{calculate}}\::\:\boldsymbol{\mathrm{I}}\:=\int_{\frac{\boldsymbol{\pi}}{\mathrm{6}}} ^{\:\frac{\boldsymbol{\pi}}{\mathrm{3}}} \boldsymbol{{ln}}\left(\boldsymbol{{tan}}\:\boldsymbol{{x}}\right)\boldsymbol{{dx}}. \\ $$$$\mathrm{2}.\:\boldsymbol{\mathrm{calculate}}\:\::\:\underset{\boldsymbol{{x}}\:\rightarrow\:\boldsymbol{{e}}} {\boldsymbol{{lim}}}\:\frac{\boldsymbol{{x}}\sqrt{\mathrm{1}−\boldsymbol{{ln}}\:\boldsymbol{{x}}}}{\boldsymbol{{x}}−\boldsymbol{{e}}}\:. \\ $$

Question Number 144001 Answers: 2 Comments: 0

Question Number 143987 Answers: 1 Comments: 0

If f(x^2 −6x+6)+f(x^2 −4x+4)=2x ∀x∈R then f(−3)+f(9)−5f(1)=?

$${If}\:{f}\left({x}^{\mathrm{2}} −\mathrm{6}{x}+\mathrm{6}\right)+{f}\left({x}^{\mathrm{2}} −\mathrm{4}{x}+\mathrm{4}\right)=\mathrm{2}{x} \\ $$$$\forall{x}\in{R}\:{th}\mathrm{e}{n}\:{f}\left(−\mathrm{3}\right)+{f}\left(\mathrm{9}\right)−\mathrm{5}{f}\left(\mathrm{1}\right)=? \\ $$

Question Number 143860 Answers: 1 Comments: 0

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

$$\mathrm{calculate}\:\int_{\mathrm{0}} ^{\infty} \:\mathrm{x}\:\mathrm{e}^{−\mathrm{x}^{\mathrm{2}} } \mathrm{log}\left(\mathrm{1}+\mathrm{e}^{\mathrm{x}} \right)\mathrm{dx} \\ $$

Question Number 143846 Answers: 1 Comments: 0

find the sum (((0!)^2 )/(1!)) +(((1!)^2 )/(3!)) +(((2!)^2 )/(5!)) +.....

$$\mathrm{find}\:\mathrm{the}\:\mathrm{sum}\:\:\frac{\left(\mathrm{0}!\right)^{\mathrm{2}} }{\mathrm{1}!}\:+\frac{\left(\mathrm{1}!\right)^{\mathrm{2}} }{\mathrm{3}!}\:+\frac{\left(\mathrm{2}!\right)^{\mathrm{2}} }{\mathrm{5}!}\:+..... \\ $$

Question Number 143730 Answers: 2 Comments: 0

find lim_(x→0) ((sin(sin(1−cosx))−1+cos(x−sinx))/x^3 )

$$\mathrm{find}\:\mathrm{lim}_{\mathrm{x}\rightarrow\mathrm{0}} \:\:\frac{\mathrm{sin}\left(\mathrm{sin}\left(\mathrm{1}−\mathrm{cosx}\right)\right)−\mathrm{1}+\mathrm{cos}\left(\mathrm{x}−\mathrm{sinx}\right)}{\mathrm{x}^{\mathrm{3}} } \\ $$

Question Number 143702 Answers: 2 Comments: 0

n ∈ IN. I_n = ∫_1 ^( e) x^(n+1) lnx dx. 1. prove that (I_n ) is positive and increasing. 2. using a part−by−part integration, calculate I_n .

Question Number 143606 Answers: 1 Comments: 0

Question Number 143576 Answers: 0 Comments: 0

find L(((arctanx)/x))

$${find}\:{L}\left(\frac{{arctanx}}{{x}}\right) \\ $$

Question Number 143575 Answers: 1 Comments: 0

find L(e^(−(√x)) )

$${find}\:{L}\left({e}^{−\sqrt{{x}}} \right) \\ $$

Pg 5 Pg 6 Pg 7 Pg 8 Pg 9 Pg 10 Pg 11 Pg 12 Pg 13 Pg 14

Contact: info@tinkutara.com