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IntegrationQuestion and Answers: Page 104

Question Number 130534 Answers: 1 Comments: 0

let f(x)=x^7 arctan(2x) 1)calculate f^((4)) (0) and f^((7)) (0) 2) calculate f^((5)) (1)

$$\mathrm{let}\:\mathrm{f}\left(\mathrm{x}\right)=\mathrm{x}^{\mathrm{7}} \:\mathrm{arctan}\left(\mathrm{2x}\right) \\ $$$$\left.\mathrm{1}\right)\mathrm{calculate}\:\:\mathrm{f}^{\left(\mathrm{4}\right)} \left(\mathrm{0}\right)\:\mathrm{and}\:\mathrm{f}^{\left(\mathrm{7}\right)} \left(\mathrm{0}\right) \\ $$$$\left.\mathrm{2}\right)\:\mathrm{calculate}\:\mathrm{f}^{\left(\mathrm{5}\right)} \left(\mathrm{1}\right) \\ $$

Question Number 130533 Answers: 1 Comments: 0

find ∫ ((x^3 +5)/(x^4 +2x^2 −3))dx

$$\mathrm{find}\:\int\:\:\frac{\mathrm{x}^{\mathrm{3}} +\mathrm{5}}{\mathrm{x}^{\mathrm{4}} +\mathrm{2x}^{\mathrm{2}} −\mathrm{3}}\mathrm{dx} \\ $$

Question Number 130532 Answers: 0 Comments: 0

find f(α)=∫_0 ^∞ ((arctan(1+αx))/(4+x^2 ))dx (α>0) and determine the value of ∫_0 ^∞ ((arctan(1+2x))/(x^2 +4))dx

$$\mathrm{find}\:\mathrm{f}\left(\alpha\right)=\int_{\mathrm{0}} ^{\infty} \:\frac{\mathrm{arctan}\left(\mathrm{1}+\alpha\mathrm{x}\right)}{\mathrm{4}+\mathrm{x}^{\mathrm{2}} }\mathrm{dx}\:\:\:\left(\alpha>\mathrm{0}\right) \\ $$$$\mathrm{and}\:\mathrm{determine}\:\mathrm{the}\:\mathrm{value}\:\mathrm{of}\:\int_{\mathrm{0}} ^{\infty} \:\frac{\mathrm{arctan}\left(\mathrm{1}+\mathrm{2x}\right)}{\mathrm{x}^{\mathrm{2}} \:+\mathrm{4}}\mathrm{dx} \\ $$

Question Number 130531 Answers: 1 Comments: 0

find ∫_0 ^1 arctan(x^2 +x+1)dx

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

Question Number 130530 Answers: 0 Comments: 2

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

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

Question Number 130529 Answers: 1 Comments: 0

calvulste ∫_0 ^(2π) ((cos(2x))/(3+cosx))dx

$$\mathrm{calvulste}\:\int_{\mathrm{0}} ^{\mathrm{2}\pi} \:\frac{\mathrm{cos}\left(\mathrm{2x}\right)}{\mathrm{3}+\mathrm{cosx}}\mathrm{dx} \\ $$

Question Number 130528 Answers: 1 Comments: 0

calculate ∫_2 ^∞ (dx/((x^2 −1)^5 ))

$$\mathrm{calculate}\:\int_{\mathrm{2}} ^{\infty} \:\:\:\:\frac{\mathrm{dx}}{\left(\mathrm{x}^{\mathrm{2}} −\mathrm{1}\right)^{\mathrm{5}} } \\ $$

Question Number 130512 Answers: 2 Comments: 2

∫xe^(1/(2x)) dx=?

$$\int{x}\mathrm{e}^{\frac{\mathrm{1}}{\mathrm{2}{x}}} {dx}=? \\ $$

Question Number 130536 Answers: 2 Comments: 0

calculate ∫_0 ^(2π) ln(x^2 −2xcosθ +1)dθ

$$\mathrm{calculate}\:\int_{\mathrm{0}} ^{\mathrm{2}\pi} \mathrm{ln}\left(\mathrm{x}^{\mathrm{2}} −\mathrm{2xcos}\theta\:+\mathrm{1}\right)\mathrm{d}\theta \\ $$

Question Number 130486 Answers: 1 Comments: 0

∫_0 ^( x) ((cos t ((sin^3 t))^(1/4) )/((sin x−sin t)^(3/4) )) dt ?

$$\:\int_{\mathrm{0}} ^{\:{x}} \:\frac{\mathrm{cos}\:{t}\:\sqrt[{\mathrm{4}}]{\mathrm{sin}^{\mathrm{3}} \:{t}}}{\left(\mathrm{sin}\:{x}−\mathrm{sin}\:{t}\right)^{\mathrm{3}/\mathrm{4}} }\:{dt}\:? \\ $$

Question Number 130485 Answers: 2 Comments: 0

ϝ = ∫_0 ^( ∞) x^(5 ) ln (x)cos (x)e^(−x) dx ?

$$\:\digamma\:=\:\int_{\mathrm{0}} ^{\:\infty} {x}^{\mathrm{5}\:} \mathrm{ln}\:\left({x}\right)\mathrm{cos}\:\left({x}\right){e}^{−{x}} \:{dx}\:? \\ $$

Question Number 130478 Answers: 1 Comments: 0

Integrate the function f(x,y)=xy(x^2 +y^2 ) over the domain R={−3≤x^2 −y^2 ≤3, 1≤y≤4 }

$$\:{Integrate}\:{the}\:{function}\:{f}\left({x},{y}\right)={xy}\left({x}^{\mathrm{2}} +{y}^{\mathrm{2}} \right) \\ $$$${over}\:{the}\:{domain}\:{R}=\left\{−\mathrm{3}\leqslant{x}^{\mathrm{2}} −{y}^{\mathrm{2}} \leqslant\mathrm{3},\:\mathrm{1}\leqslant{y}\leqslant\mathrm{4}\:\right\} \\ $$

Question Number 130474 Answers: 1 Comments: 0

E = ∫_( 0) ^( ((√3)/2)) (dy/((1−y^2 )^(5/2) ))

$$\mathcal{E}\:=\:\int_{\:\mathrm{0}} ^{\:\frac{\sqrt{\mathrm{3}}}{\mathrm{2}}} \:\frac{\mathrm{dy}}{\left(\mathrm{1}−\mathrm{y}^{\mathrm{2}} \right)^{\mathrm{5}/\mathrm{2}} }\: \\ $$

Question Number 130463 Answers: 0 Comments: 0

Question Number 130448 Answers: 1 Comments: 0

Given that f(x) = f(π−x), prove that ∫_0 ^π xf(x)dx = (π/2)∫_0 ^π f(x)dx. please what are different methods to approach this question?

$$\mathrm{Given}\:\mathrm{that}\:{f}\left({x}\right)\:=\:{f}\left(\pi−{x}\right),\:\mathrm{prove}\:\mathrm{that}\:\int_{\mathrm{0}} ^{\pi} {xf}\left({x}\right){dx}\:=\:\frac{\pi}{\mathrm{2}}\int_{\mathrm{0}} ^{\pi} {f}\left({x}\right){dx}. \\ $$$$\mathrm{please}\:\mathrm{what}\:\mathrm{are}\:\mathrm{different}\:\mathrm{methods}\:\mathrm{to}\:\mathrm{approach}\:\mathrm{this}\:\mathrm{question}? \\ $$

Question Number 130433 Answers: 2 Comments: 0

....nice calculus... prove that:: Ψ=∫_0 ^( ∞) e^(−x) ln(x)cos(x)dx=^? (1/8)(−4γ−π−2ln(2))

$$\:\:\:\:\:\:....{nice}\:\:\:{calculus}... \\ $$$$\:\:\:{prove}\:{that}:: \\ $$$$\:\:\:\Psi=\int_{\mathrm{0}} ^{\:\infty} {e}^{−{x}} {ln}\left({x}\right){cos}\left({x}\right){dx}\overset{?} {=}\frac{\mathrm{1}}{\mathrm{8}}\left(−\mathrm{4}\gamma−\pi−\mathrm{2}{ln}\left(\mathrm{2}\right)\right) \\ $$$$ \\ $$

Question Number 130432 Answers: 1 Comments: 0

...nice calculus... calculate :: Ω=∫_0 ^( ∞) xln(x)e^(−x) sin(x)dx=?

$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:...{nice}\:\:{calculus}... \\ $$$$\:\:\:\:\:{calculate}\::: \\ $$$$\:\:\:\:\Omega=\int_{\mathrm{0}} ^{\:\infty} {xln}\left({x}\right){e}^{−{x}} {sin}\left({x}\right){dx}=? \\ $$$$\:\:\: \\ $$

Question Number 130431 Answers: 2 Comments: 0

... nice calculus... please evaluate :: φ=∫_0 ^( ∞) tanh(x).e^(−sx) dx=?? ( s>0 and real...)

$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:...\:{nice}\:\:\:{calculus}... \\ $$$$\:{please}\:\:{evaluate}\::: \\ $$$$\:\:\:\phi=\int_{\mathrm{0}} ^{\:\infty} {tanh}\left({x}\right).{e}^{−{sx}} {dx}=?? \\ $$$$\:\:\:\:\:\:\:\left(\:\:{s}>\mathrm{0}\:\:\:{and}\:\:\:{real}...\right) \\ $$

Question Number 130417 Answers: 1 Comments: 0

∫_0 ^( ∞) ((sin(αx)sin(βx))/x^2 )dx

$$\int_{\mathrm{0}} ^{\:\infty} \frac{\mathrm{sin}\left(\alpha\mathrm{x}\right)\mathrm{sin}\left(\beta\mathrm{x}\right)}{\mathrm{x}^{\mathrm{2}} }\mathrm{dx} \\ $$

Question Number 130402 Answers: 0 Comments: 0

∫ (e^x ).(((x^6 +x^5 +5x^4 )/((1+x)^6 )))dx = ...

$$\: \\ $$$$\:\int\:\left(\boldsymbol{\mathrm{e}}^{\boldsymbol{\mathrm{x}}} \right).\left(\frac{\boldsymbol{\mathrm{x}}^{\mathrm{6}} +\boldsymbol{\mathrm{x}}^{\mathrm{5}} +\mathrm{5}\boldsymbol{\mathrm{x}}^{\mathrm{4}} }{\left(\mathrm{1}+\boldsymbol{\mathrm{x}}\right)^{\mathrm{6}} }\right)\boldsymbol{\mathrm{dx}}\:=\:... \\ $$

Question Number 130392 Answers: 2 Comments: 0

find ∫_(∣z∣=1) ((1−cosz)/z^2 )dz

$$\mathrm{find}\:\int_{\mid\mathrm{z}\mid=\mathrm{1}} \:\:\frac{\mathrm{1}−\mathrm{cosz}}{\mathrm{z}^{\mathrm{2}} }\mathrm{dz} \\ $$

Question Number 130391 Answers: 0 Comments: 0

calculate ∫_(∣z∣=1) ((tanz)/z)dz

$$\mathrm{calculate}\:\int_{\mid\mathrm{z}\mid=\mathrm{1}} \:\:\:\frac{\mathrm{tanz}}{\mathrm{z}}\mathrm{dz} \\ $$

Question Number 130390 Answers: 0 Comments: 0

find ∫_(∣z∣=1) (z^2 /(3+sinz))dz

$$\mathrm{find}\:\int_{\mid\mathrm{z}\mid=\mathrm{1}} \:\:\:\frac{\mathrm{z}^{\mathrm{2}} }{\mathrm{3}+\mathrm{sinz}}\mathrm{dz} \\ $$

Question Number 130389 Answers: 0 Comments: 0

let f(z)=(1/(1−cos(2z))) determine Res(f,o)

$$\mathrm{let}\:\mathrm{f}\left(\mathrm{z}\right)=\frac{\mathrm{1}}{\mathrm{1}−\mathrm{cos}\left(\mathrm{2z}\right)}\:\mathrm{determine}\:\mathrm{Res}\left(\mathrm{f},\mathrm{o}\right) \\ $$

Question Number 130387 Answers: 0 Comments: 2

calculate ∫_0 ^∞ ((cos(abx))/((x^2 +ax+1)(x^2 +bx+1))) with a and b real and ∣a∣<2,∣b∣<2

$${calculate}\:\int_{\mathrm{0}} ^{\infty} \:\frac{{cos}\left({abx}\right)}{\left({x}^{\mathrm{2}} +{ax}+\mathrm{1}\right)\left({x}^{\mathrm{2}} +{bx}+\mathrm{1}\right)} \\ $$$${with}\:{a}\:{and}\:{b}\:{real}\:{and}\:\mid{a}\mid<\mathrm{2},\mid{b}\mid<\mathrm{2} \\ $$

Question Number 130326 Answers: 2 Comments: 0

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