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IntegrationQuestion and Answers: Page 223

Question Number 67708 Answers: 1 Comments: 0

Question Number 67698 Answers: 0 Comments: 0

Question Number 67696 Answers: 0 Comments: 0

∫_0 ^1 (1−x)/(1+x)lnx dx

$$\int_{\mathrm{0}} ^{\mathrm{1}} \left(\mathrm{1}−{x}\right)/\left(\mathrm{1}+{x}\right){lnx}\:{dx} \\ $$

Question Number 67674 Answers: 0 Comments: 3

let f(a) =∫_0 ^∞ (dx/((x^2 +1)(x^2 +a))) with a>0 1) determine a explicit form of f(a) 2) calculate g(a) =∫_0 ^∞ (dx/((x^2 +1)(x^2 +a)^2 )) 3)give f^((n)) (a) at form of integral 4)calculate ∫_0 ^∞ (dx/((x^2 +1)(x^2 +3)^2 )) and ∫_0 ^∞ (dx/((x^2 +1)^3 ))

$${let}\:{f}\left({a}\right)\:=\int_{\mathrm{0}} ^{\infty} \:\:\:\:\:\frac{{dx}}{\left({x}^{\mathrm{2}} +\mathrm{1}\right)\left({x}^{\mathrm{2}} +{a}\right)}\:\:{with}\:{a}>\mathrm{0} \\ $$$$\left.\mathrm{1}\right)\:{determine}\:{a}\:{explicit}\:{form}\:{of}\:{f}\left({a}\right) \\ $$$$\left.\mathrm{2}\right)\:{calculate}\:{g}\left({a}\right)\:=\int_{\mathrm{0}} ^{\infty} \:\:\frac{{dx}}{\left({x}^{\mathrm{2}} \:+\mathrm{1}\right)\left({x}^{\mathrm{2}} \:+{a}\right)^{\mathrm{2}} } \\ $$$$\left.\mathrm{3}\right){give}\:{f}^{\left({n}\right)} \left({a}\right)\:{at}\:{form}\:{of}\:{integral} \\ $$$$\left.\mathrm{4}\right){calculate}\:\int_{\mathrm{0}} ^{\infty} \:\:\frac{{dx}}{\left({x}^{\mathrm{2}} \:+\mathrm{1}\right)\left({x}^{\mathrm{2}} \:+\mathrm{3}\right)^{\mathrm{2}} }\:{and} \\ $$$$\int_{\mathrm{0}} ^{\infty} \:\:\:\frac{{dx}}{\left({x}^{\mathrm{2}} \:+\mathrm{1}\right)^{\mathrm{3}} } \\ $$

Question Number 67673 Answers: 0 Comments: 1

calculate ∫_(−∞) ^(+∞) (dx/((x^2 −x+1)(x^2 +x+1)))

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

Question Number 67628 Answers: 0 Comments: 0

∫x^n lnx/n^(x ) dx

$$\int{x}^{{n}} {lnx}/{n}^{{x}\:} \:{dx} \\ $$

Question Number 67618 Answers: 1 Comments: 0

find the area abovnded r=cos2θ

$${find}\:{the}\:{area}\:{abovnded}\:{r}={cos}\mathrm{2}\theta \\ $$

Question Number 67617 Answers: 0 Comments: 2

Question Number 67572 Answers: 0 Comments: 1

∫_(−(π/2)) ^(π/2) {sin∣x∣+cos∣x∣} dx

$$\int_{−\frac{\pi}{\mathrm{2}}} ^{\frac{\pi}{\mathrm{2}}} \left\{{sin}\mid{x}\mid+{cos}\mid{x}\mid\right\}\:{dx} \\ $$

Question Number 67542 Answers: 0 Comments: 4

let f(a) =∫_(−∞) ^(+∞) (dx/((x^2 +1)(a +e^(ix) ))) with a>0 1)find a explicit form of f(a) 2) determine also g(a)=∫_(−∞) ^(+∞) (dx/((x^2 +1)(a+e^(ix) )^2 )) 3)let I =Re(∫_(−∞) ^(+∞) (dx/((x^2 +1)(2+e^(ix) )))) and J=Im(∫_(−∞) ^(+∞) (dx/((x^2 +1)(2+e^x )))) determine I and J and its values.

$${let}\:{f}\left({a}\right)\:=\int_{−\infty} ^{+\infty} \:\:\frac{{dx}}{\left({x}^{\mathrm{2}} +\mathrm{1}\right)\left({a}\:+{e}^{{ix}} \right)}\:\:\:{with}\:{a}>\mathrm{0} \\ $$$$\left.\mathrm{1}\right){find}\:{a}\:{explicit}\:{form}\:{of}\:{f}\left({a}\right) \\ $$$$\left.\mathrm{2}\right)\:{determine}\:{also}\:{g}\left({a}\right)=\int_{−\infty} ^{+\infty} \:\:\:\:\frac{{dx}}{\left({x}^{\mathrm{2}} +\mathrm{1}\right)\left({a}+{e}^{{ix}} \right)^{\mathrm{2}} } \\ $$$$\left.\mathrm{3}\right){let}\:{I}\:={Re}\left(\int_{−\infty} ^{+\infty} \:\:\frac{{dx}}{\left({x}^{\mathrm{2}} +\mathrm{1}\right)\left(\mathrm{2}+{e}^{{ix}} \right)}\right)\:{and}\:{J}={Im}\left(\int_{−\infty} ^{+\infty} \:\frac{{dx}}{\left({x}^{\mathrm{2}} +\mathrm{1}\right)\left(\mathrm{2}+{e}^{{x}} \right)}\right) \\ $$$$\:{determine}\:{I}\:{and}\:{J}\:\:{and}\:\:{its}\:{values}. \\ $$

Question Number 67539 Answers: 0 Comments: 3

calculate ∫_0 ^∞ (du/(∣u+z∣^2 )) if z =r e^(iθ) and −π<θ<π

$${calculate}\:\int_{\mathrm{0}} ^{\infty} \:\:\frac{{du}}{\mid{u}+{z}\mid^{\mathrm{2}} }\:\:{if}\:{z}\:={r}\:{e}^{{i}\theta} \:\:\:{and}\:−\pi<\theta<\pi \\ $$

Question Number 67531 Answers: 0 Comments: 1

prove that cos(πz) =Π_(n=1) ^∞ (1−(z^2 /(((1/2)+n)^2 )))

$${prove}\:{that}\:{cos}\left(\pi{z}\right)\:=\prod_{{n}=\mathrm{1}} ^{\infty} \left(\mathrm{1}−\frac{{z}^{\mathrm{2}} }{\left(\frac{\mathrm{1}}{\mathrm{2}}+{n}\right)^{\mathrm{2}} }\right) \\ $$

Question Number 67530 Answers: 0 Comments: 2

calculate ∫_0 ^∞ (x^(n−3) /(1+x^(2n) ))dx with n≥3

$${calculate}\:\int_{\mathrm{0}} ^{\infty} \:\:\:\frac{{x}^{{n}−\mathrm{3}} }{\mathrm{1}+{x}^{\mathrm{2}{n}} }{dx}\:\:{with}\:{n}\geqslant\mathrm{3} \\ $$

Question Number 67528 Answers: 0 Comments: 3

calculate ∫_(−∞) ^(+∞) ((1+x^3 )/(1+x^6 ))dx

$${calculate}\:\int_{−\infty} ^{+\infty} \:\:\:\frac{\mathrm{1}+{x}^{\mathrm{3}} }{\mathrm{1}+{x}^{\mathrm{6}} }{dx} \\ $$

Question Number 67527 Answers: 0 Comments: 1

calculate ∫_(−∞) ^(+∞) ((1+x^2 )/(1+x^4 ))dx

$${calculate}\:\int_{−\infty} ^{+\infty} \:\:\frac{\mathrm{1}+{x}^{\mathrm{2}} }{\mathrm{1}+{x}^{\mathrm{4}} }{dx} \\ $$$$ \\ $$

Question Number 67526 Answers: 0 Comments: 1

find the value of ∫_0 ^(2π) (dx/(3+2sinx +cosx))

$${find}\:{the}\:{value}\:{of}\:\int_{\mathrm{0}} ^{\mathrm{2}\pi} \:\:\:\frac{{dx}}{\mathrm{3}+\mathrm{2}{sinx}\:+{cosx}} \\ $$

Question Number 67525 Answers: 0 Comments: 3

let a>b>0 calculate ∫_0 ^(2π) (dx/((a+bsinx)^2 ))

$${let}\:{a}>{b}>\mathrm{0}\:{calculate}\:\int_{\mathrm{0}} ^{\mathrm{2}\pi} \:\:\frac{{dx}}{\left({a}+{bsinx}\right)^{\mathrm{2}} } \\ $$

Question Number 67517 Answers: 0 Comments: 0

let z ∈C and ∣z∣<1 prove that (z/(1−z^2 )) +(z^2 /(1−z^4 )) +.....+(z^2^n /(1−z^2^(n+1) ))+...=(z/(1−z)) (z/(1+z)) +((2z^2 )/(1+z^2 )) +....+((2^n z^2^n )/(1+z^2^n )) +....=(z/(1−z))

$${let}\:{z}\:\in{C}\:{and}\:\mid{z}\mid<\mathrm{1}\:\:{prove}\:{that} \\ $$$$\frac{{z}}{\mathrm{1}−{z}^{\mathrm{2}} }\:+\frac{{z}^{\mathrm{2}} }{\mathrm{1}−{z}^{\mathrm{4}} }\:+.....+\frac{{z}^{\mathrm{2}^{{n}} } }{\mathrm{1}−{z}^{\mathrm{2}^{{n}+\mathrm{1}} } }+...=\frac{{z}}{\mathrm{1}−{z}} \\ $$$$\frac{{z}}{\mathrm{1}+{z}}\:+\frac{\mathrm{2}{z}^{\mathrm{2}} }{\mathrm{1}+{z}^{\mathrm{2}} }\:+....+\frac{\mathrm{2}^{{n}} \:{z}^{\mathrm{2}^{{n}} } }{\mathrm{1}+\mathrm{z}^{\mathrm{2}^{\mathrm{n}} } }\:+....=\frac{\mathrm{z}}{\mathrm{1}−\mathrm{z}} \\ $$

Question Number 67513 Answers: 0 Comments: 0

∫x^(n ) lnx/n^x dx

$$\int{x}^{{n}\:} {lnx}/{n}^{{x}} \:{dx} \\ $$

Question Number 67467 Answers: 0 Comments: 0

Find f(x)=∫_0 ^∞ (( tlnt)/((1+t^2 )^x )) dt

$$ \\ $$$$ \\ $$$${Find}\:\:\:\:{f}\left({x}\right)=\int_{\mathrm{0}} ^{\infty} \:\:\frac{\:{tlnt}}{\left(\mathrm{1}+{t}^{\mathrm{2}} \right)^{{x}} }\:{dt}\: \\ $$

Question Number 67466 Answers: 0 Comments: 0

let consider for all n≥1 the real (t)_n =t(t+1).....(t+n−1) Find L_n = ∫_0 ^∞ (((t)_1 )/((t)_(n+1) )) dt

$$ \\ $$$$ \\ $$$${let}\:{consider}\:\:\:{for}\:{all}\:{n}\geqslant\mathrm{1}\:{the}\:{real}\:\left({t}\right)_{{n}} \:={t}\left({t}+\mathrm{1}\right).....\left({t}+{n}−\mathrm{1}\right) \\ $$$${Find}\:\:\:{L}_{{n}} =\:\int_{\mathrm{0}} ^{\infty} \:\:\frac{\left({t}\right)_{\mathrm{1}} }{\left({t}\right)_{{n}+\mathrm{1}} }\:{dt} \\ $$

Question Number 67463 Answers: 1 Comments: 3

Find Find K=∫_0 ^(π/2) (√(tanθ)) dθ