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IntegrationQuestion and Answers: Page 267

Question Number 45641 Answers: 2 Comments: 1

Question Number 45634 Answers: 0 Comments: 0

1)find ∫ ln(1−x^6 )dx 2) calculate ∫_0 ^1 ln(1−x^6 )dx

$$\left.\mathrm{1}\right){find}\:\int\:{ln}\left(\mathrm{1}−{x}^{\mathrm{6}} \right){dx} \\ $$$$\left.\mathrm{2}\right)\:{calculate}\:\int_{\mathrm{0}} ^{\mathrm{1}} {ln}\left(\mathrm{1}−{x}^{\mathrm{6}} \right){dx} \\ $$

Question Number 45632 Answers: 0 Comments: 2

1)find ∫ ln(1+x^3 )dx 2) calculate ∫_0 ^1 ln(1+x^3 )ex

$$\left.\mathrm{1}\right){find}\:\int\:{ln}\left(\mathrm{1}+{x}^{\mathrm{3}} \right){dx} \\ $$$$\left.\mathrm{2}\right)\:{calculate}\:\int_{\mathrm{0}} ^{\mathrm{1}} {ln}\left(\mathrm{1}+{x}^{\mathrm{3}} \right){ex} \\ $$

Question Number 45600 Answers: 0 Comments: 2

find f(x,y) =∫_0 ^(π/2) ln(x+y sinθ)dθ with ∣y∣<∣x∣ 2) find f(2,3) 3)find f((√2),(√3)) .

$${find}\:{f}\left({x},{y}\right)\:=\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} {ln}\left({x}+{y}\:{sin}\theta\right){d}\theta\:\:{with}\:\:\mid{y}\mid<\mid{x}\mid \\ $$$$\left.\mathrm{2}\right)\:{find}\:{f}\left(\mathrm{2},\mathrm{3}\right) \\ $$$$\left.\mathrm{3}\right){find}\:{f}\left(\sqrt{\mathrm{2}},\sqrt{\mathrm{3}}\right)\:. \\ $$

Question Number 45561 Answers: 0 Comments: 2

Question Number 45520 Answers: 0 Comments: 1

let a>0 and b>0 calculate ∫ (√(acos^2 θ +bsin^2 θ))dπ 2) find ∫_(π/4) ^(π/2) (√(2cos^2 θ +3 sin^2 θ))dθ .

$${let}\:{a}>\mathrm{0}\:{and}\:{b}>\mathrm{0}\:{calculate}\:\int\:\sqrt{{acos}^{\mathrm{2}} \theta\:+{bsin}^{\mathrm{2}} \theta}{d}\pi \\ $$$$\left.\mathrm{2}\right)\:{find}\:\int_{\frac{\pi}{\mathrm{4}}} ^{\frac{\pi}{\mathrm{2}}} \sqrt{\mathrm{2}{cos}^{\mathrm{2}} \theta\:+\mathrm{3}\:{sin}^{\mathrm{2}} \theta}{d}\theta\:. \\ $$

Question Number 45519 Answers: 0 Comments: 2

find ∫ (√(2+tan^2 θ))dθ

$${find}\:\:\int\:\sqrt{\mathrm{2}+{tan}^{\mathrm{2}} \theta}{d}\theta \\ $$

Question Number 45498 Answers: 1 Comments: 3

Question Number 45495 Answers: 1 Comments: 0

Question Number 45482 Answers: 1 Comments: 0

Question Number 45373 Answers: 1 Comments: 1

∫(t^3 /(1+t))dt=?

$$\int\frac{\mathrm{t}^{\mathrm{3}} }{\mathrm{1}+\mathrm{t}}\mathrm{dt}=? \\ $$

Question Number 45352 Answers: 1 Comments: 0

Question Number 45334 Answers: 1 Comments: 0

Question Number 45264 Answers: 0 Comments: 4

Question Number 45235 Answers: 0 Comments: 2

let ∣a∣<1 and f(a)=∫_0 ^1 ln(x)ln(1+ax)dx 1) find a explicit form of f(a) 2) calculate g(a) =∫_0 ^1 ((xln(x))/(1+ax))dx 3) calculate ∫_0 ^1 ln(x)ln(2+x)dx 4) calculate ∫_0 ^1 ((xln(x))/(2+x))dx 5) calculate u_n =∫_0 ^1 ((xln(x))/(n+x))dx with n integr and n>1 find nature of the serie Σ u_n

$${let}\:\mid{a}\mid<\mathrm{1}\:{and}\:\:{f}\left({a}\right)=\int_{\mathrm{0}} ^{\mathrm{1}} {ln}\left({x}\right){ln}\left(\mathrm{1}+{ax}\right){dx} \\ $$$$\left.\mathrm{1}\right)\:{find}\:{a}\:{explicit}\:{form}\:{of}\:{f}\left({a}\right) \\ $$$$\left.\mathrm{2}\right)\:{calculate}\:\:{g}\left({a}\right)\:=\int_{\mathrm{0}} ^{\mathrm{1}} \:\:\frac{{xln}\left({x}\right)}{\mathrm{1}+{ax}}{dx} \\ $$$$\left.\mathrm{3}\right)\:{calculate}\:\int_{\mathrm{0}} ^{\mathrm{1}} {ln}\left({x}\right){ln}\left(\mathrm{2}+{x}\right){dx} \\ $$$$\left.\mathrm{4}\right)\:{calculate}\:\int_{\mathrm{0}} ^{\mathrm{1}} \:\:\:\frac{{xln}\left({x}\right)}{\mathrm{2}+{x}}{dx}\: \\ $$$$\left.\mathrm{5}\right)\:{calculate}\:{u}_{{n}} =\int_{\mathrm{0}} ^{\mathrm{1}} \:\:\frac{{xln}\left({x}\right)}{{n}+{x}}{dx}\:{with}\:{n}\:{integr}\:{and}\:{n}>\mathrm{1} \\ $$$${find}\:{nature}\:{of}\:{the}\:{serie}\:\Sigma\:{u}_{{n}} \\ $$

Question Number 45201 Answers: 0 Comments: 6

find ∫ (dx/(√(1+x^3 )))

$${find}\:\int\:\:\frac{{dx}}{\sqrt{\mathrm{1}+{x}^{\mathrm{3}} }} \\ $$

Question Number 45232 Answers: 1 Comments: 1

calculate ∫_0 ^1 ln(x)ln(1+x)dx .

$${calculate}\:\:\int_{\mathrm{0}} ^{\mathrm{1}} {ln}\left({x}\right){ln}\left(\mathrm{1}+{x}\right){dx}\:. \\ $$

Question Number 45231 Answers: 2 Comments: 1

find ∫ (√((x−1)(3−x)))dx

$${find}\:\int\:\sqrt{\left({x}−\mathrm{1}\right)\left(\mathrm{3}−{x}\right)}{dx} \\ $$

Question Number 45155 Answers: 2 Comments: 1