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IntegrationQuestion and Answers: Page 100

Question Number 136626 Answers: 1 Comments: 0

..........nice calculus......... suppose that::: ϕ(p)=∫_0 ^( ∞) ((ln(1+x))/((p+x)^2 )) ...✓ find the value of:: ∫^( 1) _0 ((ϕ(p))/(1+p))dp=?...

$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:..........{nice}\:\:\:\:{calculus}......... \\ $$$$\:\:\:\:{suppose}\:{that}::: \\ $$$$\:\:\:\:\:\:\:\:\:\:\varphi\left({p}\right)=\int_{\mathrm{0}} ^{\:\infty} \frac{{ln}\left(\mathrm{1}+{x}\right)}{\left({p}+{x}\right)^{\mathrm{2}} }\:...\checkmark \\ $$$$\:\:\:\:\:\:\:\:\:\:{find}\:{the}\:{value}\:{of}:: \\ $$$$\:\:\:\:\:\:\:\underset{\mathrm{0}} {\int}^{\:\mathrm{1}} \:\frac{\varphi\left({p}\right)}{\mathrm{1}+{p}}{dp}=?... \\ $$$$\:\:\:\:\:\: \\ $$

Question Number 136572 Answers: 3 Comments: 1

....advanced calculus.... 𝛗=∫_0 ^( (π/2)) x.(tan(x))^(1/2) dx=??

$$\:\:\:\:\:\:\:\:\:\:\:\:\:....{advanced}\:\:\:\:{calculus}.... \\ $$$$\:\:\:\:\:\:\:\:\boldsymbol{\phi}=\int_{\mathrm{0}} ^{\:\frac{\pi}{\mathrm{2}}} {x}.\left({tan}\left({x}\right)\right)^{\frac{\mathrm{1}}{\mathrm{2}}} {dx}=?? \\ $$$$ \\ $$

Question Number 136497 Answers: 1 Comments: 2

......nice calculus..... prove:: ∫_0 ^( 1) (1/(1+ln^2 (x)))dx=∫_(0 ) ^( ∞) ((sin(x))/(1+x))dx

$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:......{nice}\:\:\:\:{calculus}..... \\ $$$$\:\:\:\:{prove}::\:\: \\ $$$$\:\:\:\:\:\:\:\int_{\mathrm{0}} ^{\:\mathrm{1}} \frac{\mathrm{1}}{\mathrm{1}+{ln}^{\mathrm{2}} \left({x}\right)}{dx}=\int_{\mathrm{0}\:} ^{\:\infty} \frac{{sin}\left({x}\right)}{\mathrm{1}+{x}}{dx} \\ $$$$ \\ $$

Question Number 136482 Answers: 0 Comments: 0

f(x)=∫_(−Π/4) ^(Π∫/4) e^(xtant) dt

$${f}\left({x}\right)=\int_{−\Pi/\mathrm{4}} ^{\Pi\int/\mathrm{4}} {e}^{{xtant}} {dt} \\ $$

Question Number 136481 Answers: 1 Comments: 0

∫_0 ^((50π)/3) ∣sinx∣dx

$$\int_{\mathrm{0}} ^{\frac{\mathrm{50}\pi}{\mathrm{3}}} \mid{sinx}\mid{dx} \\ $$

Question Number 136476 Answers: 1 Comments: 0

(a) Let I(α)=∫_0 ^∞ e^(−(x−(α/x))^2 ) dx Show that it is legitimate to take the derivative of I(α) and also I′(α)= 0. Then show that I(α)=((√π)/2). (b) Use (a) to prove ∫_0 ^∞ e^(−(x^2 +α^2 x^(−2) )) dx=((√π)/2)e^(−2α) .

$$\left(\mathrm{a}\right)\:\mathrm{Let}\: \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\mathrm{I}\left(\alpha\right)=\int_{\mathrm{0}} ^{\infty} \mathrm{e}^{−\left(\mathrm{x}−\frac{\alpha}{\mathrm{x}}\right)^{\mathrm{2}} } \mathrm{dx} \\ $$$$\mathrm{Show}\:\mathrm{that}\:\mathrm{it}\:\mathrm{is}\:\mathrm{legitimate}\:\mathrm{to}\:\mathrm{take}\:\mathrm{the}\:\mathrm{derivative}\:\mathrm{of}\:\mathrm{I}\left(\alpha\right)\:\mathrm{and}\:\mathrm{also}\:\mathrm{I}'\left(\alpha\right)= \\ $$$$\mathrm{0}.\:\mathrm{Then}\:\mathrm{show}\:\mathrm{that} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\mathrm{I}\left(\alpha\right)=\frac{\sqrt{\pi}}{\mathrm{2}}. \\ $$$$\left(\mathrm{b}\right)\:\mathrm{Use}\:\left(\mathrm{a}\right)\:\mathrm{to}\:\mathrm{prove} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\int_{\mathrm{0}} ^{\infty} \mathrm{e}^{−\left(\mathrm{x}^{\mathrm{2}} +\alpha^{\mathrm{2}} \mathrm{x}^{−\mathrm{2}} \right)} \mathrm{dx}=\frac{\sqrt{\pi}}{\mathrm{2}}\mathrm{e}^{−\mathrm{2}\alpha} . \\ $$$$ \\ $$

Question Number 136473 Answers: 0 Comments: 0

Question Number 136445 Answers: 2 Comments: 0

Question Number 136440 Answers: 2 Comments: 0

∫ (dx/(sin^6 x)) ?

$$\:\int\:\frac{{dx}}{\mathrm{sin}\:^{\mathrm{6}} {x}}\:? \\ $$

Question Number 136425 Answers: 1 Comments: 3

If α>0 and β>0, prove ∫_0 ^∞ ((ln(αx))/(β^2 +x^2 ))dx=(π/(2β))ln(αβ)

$$\mathrm{If}\:\alpha>\mathrm{0}\:\mathrm{and}\:\beta>\mathrm{0},\:\mathrm{prove} \\ $$$$\int_{\mathrm{0}} ^{\infty} \frac{\mathrm{ln}\left(\alpha\mathrm{x}\right)}{\beta^{\mathrm{2}} +\mathrm{x}^{\mathrm{2}} }\mathrm{dx}=\frac{\pi}{\mathrm{2}\beta}\mathrm{ln}\left(\alpha\beta\right) \\ $$

Question Number 136406 Answers: 0 Comments: 1

calculate A_λ =∫_0 ^∞ ((cos^4 x)/((x^2 +λ^2 )^2 ))dx 2) find the value of ∫_0 ^∞ ((cos^4 x)/((x^2 +3)^2 ))dx

$$\mathrm{calculate}\:\:\mathrm{A}_{\lambda} =\int_{\mathrm{0}} ^{\infty} \:\:\frac{\mathrm{cos}^{\mathrm{4}} \mathrm{x}}{\left(\mathrm{x}^{\mathrm{2}} \:+\lambda^{\mathrm{2}} \right)^{\mathrm{2}} }\mathrm{dx} \\ $$$$\left.\mathrm{2}\right)\:\mathrm{find}\:\mathrm{the}\:\mathrm{value}\:\mathrm{of}\:\int_{\mathrm{0}} ^{\infty} \:\frac{\mathrm{cos}^{\mathrm{4}} \mathrm{x}}{\left(\mathrm{x}^{\mathrm{2}} \:+\mathrm{3}\right)^{\mathrm{2}} }\mathrm{dx} \\ $$

Question Number 136405 Answers: 0 Comments: 0

if f(x)=x^3 −3x+2 determine f^(−1) (x) and ∫ f^(−1) (nf(x))dx with n integr

$$\mathrm{if}\:\mathrm{f}\left(\mathrm{x}\right)=\mathrm{x}^{\mathrm{3}} −\mathrm{3x}+\mathrm{2}\:\:\mathrm{determine}\:\:\mathrm{f}^{−\mathrm{1}} \left(\mathrm{x}\right)\:\mathrm{and}\:\int\:\:\mathrm{f}^{−\mathrm{1}} \left(\mathrm{nf}\left(\mathrm{x}\right)\right)\mathrm{dx}\:\:\mathrm{with} \\ $$$$\mathrm{n}\:\mathrm{integr} \\ $$

Question Number 136403 Answers: 0 Comments: 0

find ∫ ((arctan(2x))/(x+3))dx

$$\mathrm{find}\:\int\:\frac{\mathrm{arctan}\left(\mathrm{2x}\right)}{\mathrm{x}+\mathrm{3}}\mathrm{dx} \\ $$

Question Number 136402 Answers: 0 Comments: 0

find U_n =∫_(1/n) ^n (1−(1/x^2 ))arctan(x+(1/x))dx and lim_(n→∞) U_n

$$\mathrm{find}\:\mathrm{U}_{\mathrm{n}} =\int_{\frac{\mathrm{1}}{\mathrm{n}}} ^{\mathrm{n}} \:\left(\mathrm{1}−\frac{\mathrm{1}}{\mathrm{x}^{\mathrm{2}} }\right)\mathrm{arctan}\left(\mathrm{x}+\frac{\mathrm{1}}{\mathrm{x}}\right)\mathrm{dx} \\ $$$$\mathrm{and}\:\mathrm{lim}_{\mathrm{n}\rightarrow\infty} \mathrm{U}_{\mathrm{n}} \\ $$

Question Number 136401 Answers: 1 Comments: 0

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

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

Question Number 136400 Answers: 0 Comments: 0

calculate ∫∫_([0,1]^2 ) e^(−(x^2 +y^2 )) arctan(x^2 +y^2 )dxdy

$$\mathrm{calculate}\:\:\int\int_{\left[\mathrm{0},\mathrm{1}\right]^{\mathrm{2}} } \:\:\mathrm{e}^{−\left(\mathrm{x}^{\mathrm{2}} +\mathrm{y}^{\mathrm{2}} \right)} \:\:\:\mathrm{arctan}\left(\mathrm{x}^{\mathrm{2}} \:+\mathrm{y}^{\mathrm{2}} \right)\mathrm{dxdy} \\ $$

Question Number 136399 Answers: 0 Comments: 0

calculate ∫ (dx/(x^n (√(x^2 −1))))

$$\mathrm{calculate}\:\int\:\:\frac{\mathrm{dx}}{\mathrm{x}^{\mathrm{n}} \sqrt{\mathrm{x}^{\mathrm{2}} −\mathrm{1}}} \\ $$

Question Number 136396 Answers: 0 Comments: 1

find ∫_0 ^1 (x^a /(1−x))dx

$$\mathrm{find}\:\int_{\mathrm{0}} ^{\mathrm{1}} \:\frac{\mathrm{x}^{\mathrm{a}} }{\mathrm{1}−\mathrm{x}}\mathrm{dx} \\ $$

Question Number 136381 Answers: 1 Comments: 0

......sdvanced cslculus...... if x∈R^+ and:: 𝛗(x)=∫_0 ^( x) ((e^t −1)/t)ln((x/t))dt then prove that :: Ψ=∫_0 ^( ∞) e^(−x) 𝛗(x)dx=ζ(2)

$$\:\:\:\:\:\:\:\:\:\:\:\:\:......{sdvanced}\:\:\:{cslculus}...... \\ $$$$\:{if}\:\:{x}\in\mathbb{R}^{+} \:{and}::\: \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\boldsymbol{\phi}\left({x}\right)=\int_{\mathrm{0}} ^{\:{x}} \frac{{e}^{{t}} −\mathrm{1}}{{t}}{ln}\left(\frac{{x}}{{t}}\right){dt} \\ $$$$\:\:{then}\:{prove}\:\:{that}\::: \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\Psi=\int_{\mathrm{0}} ^{\:\infty} {e}^{−{x}} \boldsymbol{\phi}\left({x}\right){dx}=\zeta\left(\mathrm{2}\right) \\ $$

Question Number 136365 Answers: 3 Comments: 0

∫ (dx/(sin x (√(cos x)))) =?

$$\int\:\frac{{dx}}{\mathrm{sin}\:{x}\:\sqrt{\mathrm{cos}\:{x}}}\:=? \\ $$

Question Number 136343 Answers: 1 Comments: 0

lim_(x→0) (x^(2021) /(x−ln(Σ_(k=0) ^(2020) (x^k /(k!))))) =^? 2021!

$$\:\:\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\:\frac{{x}^{\mathrm{2021}} }{{x}−{ln}\left(\underset{{k}=\mathrm{0}} {\overset{\mathrm{2020}} {\sum}}\frac{{x}^{{k}} }{{k}!}\right)}\:\overset{?} {=}\:\mathrm{2021}!\: \\ $$

Question Number 136333 Answers: 0 Comments: 0

calculate ∫_0 ^∞ (t^a /(1+t+t^2 ))dt study first the convergence (a real)

$${calculate}\:\int_{\mathrm{0}} ^{\infty} \:\frac{{t}^{{a}} }{\mathrm{1}+{t}+{t}^{\mathrm{2}} }{dt} \\ $$$${study}\:{first}\:{the}\:{convergence} \\ $$$$\left({a}\:{real}\right) \\ $$

Question Number 136325 Answers: 0 Comments: 1

please a generall Form for C(n) C(n)=(4/π^2 )Σ_(k=1) ^n (−1)^(k−1) ζ(2k)ζ(2n−2k)

$${please}\:{a}\:{generall}\:{Form}\:{for}\:{C}\left({n}\right) \\ $$$${C}\left({n}\right)=\frac{\mathrm{4}}{\pi^{\mathrm{2}} }\underset{{k}=\mathrm{1}} {\overset{{n}} {\sum}}\left(−\mathrm{1}\right)^{{k}−\mathrm{1}} \zeta\left(\mathrm{2}{k}\right)\zeta\left(\mathrm{2}{n}−\mathrm{2}{k}\right) \\ $$

Question Number 136279 Answers: 1 Comments: 0

Question Number 136267 Answers: 0 Comments: 1

Question Number 136211 Answers: 1 Comments: 0

hi, guyz ! please, i need ur help ! I=∫_1 ^( 5) xe^x^3 dx

$$\boldsymbol{\mathrm{hi}},\:\boldsymbol{\mathrm{guyz}}\:!\: \\ $$$$\boldsymbol{\mathrm{please}},\:\boldsymbol{\mathrm{i}}\:\boldsymbol{\mathrm{need}}\:\boldsymbol{\mathrm{ur}}\:\boldsymbol{\mathrm{help}}\:! \\ $$$$\boldsymbol{\mathrm{I}}=\int_{\mathrm{1}} ^{\:\mathrm{5}} \boldsymbol{{xe}}^{\boldsymbol{{x}}^{\mathrm{3}} } \boldsymbol{{dx}} \\ $$

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