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IntegrationQuestion and Answers: Page 217

Question Number 65015    Answers: 1   Comments: 3

∫(((√(x+1)) − (√(x−1)))/((√(x+1)) + (√(x−1)))) dx

$$\int\frac{\sqrt{{x}+\mathrm{1}}\:−\:\sqrt{{x}−\mathrm{1}}}{\sqrt{{x}+\mathrm{1}}\:+\:\sqrt{{x}−\mathrm{1}}}\:{dx} \\ $$

Question Number 65004    Answers: 0   Comments: 1

let U_n = ∫_(1/n) ^(2/n) Γ(x)Γ(1−x)dx with n≥3 1) calculate and determine lim_(n→+∞) U_n 2) study the convergence of Σ U_n

$${let}\:{U}_{{n}} =\:\int_{\frac{\mathrm{1}}{{n}}} ^{\frac{\mathrm{2}}{{n}}} \:\Gamma\left({x}\right)\Gamma\left(\mathrm{1}−{x}\right){dx}\:\:\:\:{with}\:{n}\geqslant\mathrm{3} \\ $$$$\left.\mathrm{1}\right)\:{calculate}\:{and}\:{determine}\:{lim}_{{n}\rightarrow+\infty} \:{U}_{{n}} \\ $$$$\left.\mathrm{2}\right)\:{study}\:{the}\:{convergence}\:{of}\:\Sigma\:{U}_{{n}} \\ $$

Question Number 65003    Answers: 0   Comments: 0

find ∫_0 ^∞ (dx/(Γ(x))) with Γ(x) =∫_0 ^∞ t^(x−1) e^(−t) dt (x>0)

$${find}\:\int_{\mathrm{0}} ^{\infty} \:\:\:\frac{{dx}}{\Gamma\left({x}\right)}\:\:{with}\:\:\Gamma\left({x}\right)\:=\int_{\mathrm{0}} ^{\infty} \:{t}^{{x}−\mathrm{1}} \:{e}^{−{t}} \:{dt}\:\:\:\left({x}>\mathrm{0}\right) \\ $$

Question Number 64994    Answers: 1   Comments: 1

∫(√(tanh(x))) dx

$$\int\sqrt{{tanh}\left({x}\right)}\:{dx} \\ $$

Question Number 64993    Answers: 0   Comments: 0

let f(a) =∫_0 ^∞ ((cos(x^2 )−sin(x^2 ))/((x^2 +a^2 )^2 ))dx with a>0 1) calculate f(a) 2) find the values of ∫_0 ^∞ ((cos(x^2 )−sin(x^2 ))/((x^2 +1)^2 )) and ∫_0 ^∞ ((cos(x^2 )−sin(x^2 ))/((x^2 +3)^2 ))dx .

$${let}\:{f}\left({a}\right)\:=\int_{\mathrm{0}} ^{\infty} \:\:\frac{{cos}\left({x}^{\mathrm{2}} \right)−{sin}\left({x}^{\mathrm{2}} \right)}{\left({x}^{\mathrm{2}} +{a}^{\mathrm{2}} \right)^{\mathrm{2}} }{dx}\:\:{with}\:{a}>\mathrm{0} \\ $$$$\left.\mathrm{1}\right)\:{calculate}\:{f}\left({a}\right) \\ $$$$\left.\mathrm{2}\right)\:{find}\:{the}\:{values}\:{of}\:\int_{\mathrm{0}} ^{\infty} \:\:\:\frac{{cos}\left({x}^{\mathrm{2}} \right)−{sin}\left({x}^{\mathrm{2}} \right)}{\left({x}^{\mathrm{2}} \:+\mathrm{1}\right)^{\mathrm{2}} }\:{and} \\ $$$$\int_{\mathrm{0}} ^{\infty} \:\:\frac{{cos}\left({x}^{\mathrm{2}} \right)−{sin}\left({x}^{\mathrm{2}} \right)}{\left({x}^{\mathrm{2}} \:+\mathrm{3}\right)^{\mathrm{2}} }{dx}\:. \\ $$

Question Number 64975    Answers: 0   Comments: 1

Question Number 64970    Answers: 0   Comments: 9

let f(a)=∫_0 ^∞ ((cos(x^2 ) +sin(x^2 ))/((x^2 +a^2 )^2 )) dx with a>0 1) calculate f(a) 2) find the values of ∫_0 ^∞ ((cos(x^2 )+sin(x^2 ))/((x^2 +1)^2 ))dx and ∫_0 ^∞ ((cos(x^2 )+sin(x^2 ))/((x^2 +3)^2 ))dx

$${let}\:{f}\left({a}\right)=\int_{\mathrm{0}} ^{\infty} \:\:\frac{{cos}\left({x}^{\mathrm{2}} \right)\:+{sin}\left({x}^{\mathrm{2}} \right)}{\left({x}^{\mathrm{2}} \:+{a}^{\mathrm{2}} \right)^{\mathrm{2}} }\:{dx}\:\:\:{with}\:{a}>\mathrm{0} \\ $$$$\left.\mathrm{1}\right)\:{calculate}\:{f}\left({a}\right) \\ $$$$\left.\mathrm{2}\right)\:{find}\:{the}\:{values}\:{of}\:\int_{\mathrm{0}} ^{\infty} \:\:\frac{{cos}\left({x}^{\mathrm{2}} \right)+{sin}\left({x}^{\mathrm{2}} \right)}{\left({x}^{\mathrm{2}} +\mathrm{1}\right)^{\mathrm{2}} }{dx}\:{and} \\ $$$$\int_{\mathrm{0}} ^{\infty} \:\frac{{cos}\left({x}^{\mathrm{2}} \right)+{sin}\left({x}^{\mathrm{2}} \right)}{\left({x}^{\mathrm{2}} +\mathrm{3}\right)^{\mathrm{2}} }{dx} \\ $$

Question Number 64905    Answers: 0   Comments: 0

∫log (tan x)dx

$$\int\mathrm{log}\:\left(\mathrm{tan}\:\mathrm{x}\right)\mathrm{dx} \\ $$

Question Number 64904    Answers: 0   Comments: 2

calculate ∫_1 ^2 ∫_0 ^x (1/((x^2 +y^2 )^(3/2) ))dydx

$${calculate}\:\int_{\mathrm{1}} ^{\mathrm{2}} \:\int_{\mathrm{0}} ^{{x}} \:\:\frac{\mathrm{1}}{\left({x}^{\mathrm{2}} \:+{y}^{\mathrm{2}} \right)^{\frac{\mathrm{3}}{\mathrm{2}}} }{dydx}\: \\ $$

Question Number 64866    Answers: 0   Comments: 3

find ∫_1 ^(+∞) (dx/(x^2 (√(1+x+x^2 ))))

$${find}\:\int_{\mathrm{1}} ^{+\infty} \:\frac{{dx}}{{x}^{\mathrm{2}} \sqrt{\mathrm{1}+{x}+{x}^{\mathrm{2}} }} \\ $$

Question Number 64850    Answers: 0   Comments: 4

let A_λ =∫_0 ^π (dx/(λ +cosx +sinx)) (λ ∈ R) 1) find a explicit form of A_λ 2) find also B_λ =∫_0 ^π (dx/((λ +cosx +sinx)^2 )) 3) calculate ∫_0 ^π (dx/(2+cosx +sinx)) and ∫_0 ^π (dx/((3+cosx +sinx)^2 ))

$${let}\:{A}_{\lambda} \:=\int_{\mathrm{0}} ^{\pi} \:\:\:\frac{{dx}}{\lambda\:\:+{cosx}\:+{sinx}}\:\:\:\:\left(\lambda\:\in\:{R}\right) \\ $$$$\left.\mathrm{1}\right)\:{find}\:{a}\:{explicit}\:{form}\:{of}\:{A}_{\lambda} \\ $$$$\left.\mathrm{2}\right)\:\:{find}\:{also}\:{B}_{\lambda} \:=\int_{\mathrm{0}} ^{\pi} \:\:\frac{{dx}}{\left(\lambda\:+{cosx}\:+{sinx}\right)^{\mathrm{2}} } \\ $$$$\left.\mathrm{3}\right)\:{calculate}\:\int_{\mathrm{0}} ^{\pi} \:\:\:\frac{{dx}}{\mathrm{2}+{cosx}\:+{sinx}}\:\:{and}\:\int_{\mathrm{0}} ^{\pi} \:\:\frac{{dx}}{\left(\mathrm{3}+{cosx}\:+{sinx}\right)^{\mathrm{2}} } \\ $$

Question Number 64818    Answers: 2   Comments: 1

find ∫ (dx/(1+cosx +cos(2x)))

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

Question Number 64805    Answers: 0   Comments: 0

∫log(((1+sinhx))/((1−sinhx)))tanhx dx

$$\int{log}\frac{\left(\mathrm{1}+{sinhx}\right)}{\left(\mathrm{1}−{sinhx}\right)}{tanhx}\:{dx} \\ $$

Question Number 64802    Answers: 1   Comments: 1

∫(cos^4 x+sin^4 x)/(cos2x+1)dx

$$\int\left({cos}^{\mathrm{4}} {x}+{sin}^{\mathrm{4}} {x}\right)/\left({cos}\mathrm{2}{x}+\mathrm{1}\right){dx} \\ $$

Question Number 64801    Answers: 0   Comments: 0

∫(cos^4 x+sin^4 x)/(cos2x+1)dx

$$\int\left({cos}^{\mathrm{4}} {x}+{sin}^{\mathrm{4}} {x}\right)/\left({cos}\mathrm{2}{x}+\mathrm{1}\right){dx} \\ $$

Question Number 64762    Answers: 1   Comments: 1

Given that g(x)=(2/((1+x)(1+3x^2 )) a) express g(x) in partial fractions. b) evaluate ∫_0 ^1 g((x) dx.

$$\mathrm{Given}\:\mathrm{that}\:\mathrm{g}\left(\mathrm{x}\right)=\frac{\mathrm{2}}{\left(\mathrm{1}+\mathrm{x}\right)\left(\mathrm{1}+\mathrm{3x}^{\mathrm{2}} \right.} \\ $$$$\left.\mathrm{a}\right)\:\mathrm{express}\:\mathrm{g}\left(\mathrm{x}\right)\:\mathrm{in}\:\mathrm{partial}\:\mathrm{fractions}. \\ $$$$\left.\mathrm{b}\right)\:\mathrm{evaluate}\:\int_{\mathrm{0}} ^{\mathrm{1}} \mathrm{g}\left(\left(\mathrm{x}\right)\:\mathrm{dx}.\right. \\ $$

Question Number 64759    Answers: 0   Comments: 0

∫((ln(ln(x)))/((ln(x))^n )) dx , n≠1

$$\int\frac{{ln}\left({ln}\left({x}\right)\right)}{\left({ln}\left({x}\right)\right)^{{n}} }\:{dx}\:\:\:,\:\:\:{n}\neq\mathrm{1} \\ $$

Question Number 64745    Answers: 0   Comments: 1

calculate ∫_0 ^1 ((sin(lnx))/(lnx))dx

$${calculate}\:\int_{\mathrm{0}} ^{\mathrm{1}} \:\frac{{sin}\left({lnx}\right)}{{lnx}}{dx} \\ $$

Question Number 64873    Answers: 0   Comments: 4

let f(x) =∫_0 ^π (dt/(x+sint)) with xreal 1) find a explicit form of f(x) 2) find also g(x) =∫_0 ^π (dt/((x+sint)^2 )) 3) give f^((n)) (x) at form of integral 4) calculate ∫_0 ^π (dt/(3+sint)) and ∫_0 ^π (dt/((3+sint)^2 ))

$${let}\:{f}\left({x}\right)\:=\int_{\mathrm{0}} ^{\pi} \:\:\frac{{dt}}{{x}+{sint}}\:\:\:{with}\:{xreal} \\ $$$$\left.\mathrm{1}\right)\:{find}\:{a}\:{explicit}\:{form}\:{of}\:{f}\left({x}\right) \\ $$$$\left.\mathrm{2}\right)\:{find}\:{also}\:{g}\left({x}\right)\:=\int_{\mathrm{0}} ^{\pi} \:\:\frac{{dt}}{\left({x}+{sint}\right)^{\mathrm{2}} } \\ $$$$\left.\mathrm{3}\right)\:{give}\:{f}^{\left({n}\right)} \left({x}\right)\:{at}\:{form}\:{of}\:{integral} \\ $$$$\left.\mathrm{4}\right)\:{calculate}\:\int_{\mathrm{0}} ^{\pi} \:\:\frac{{dt}}{\mathrm{3}+{sint}}\:\:{and}\:\int_{\mathrm{0}} ^{\pi} \:\:\frac{{dt}}{\left(\mathrm{3}+{sint}\right)^{\mathrm{2}} } \\ $$

Question Number 64735    Answers: 0   Comments: 2

∫tanθ/1+^− sinθ dθ

$$\int{tan}\theta/\mathrm{1}\overset{−} {+}{sin}\theta\:{d}\theta \\ $$

Question Number 64733    Answers: 0   Comments: 0

∫(secθtanθ)dθ/secθ+^− tanθ

$$\int\left({sec}\theta{tan}\theta\right){d}\theta/{sec}\theta\overset{−} {+}{tan}\theta \\ $$

Question Number 64687    Answers: 0   Comments: 1

Question Number 64677    Answers: 0   Comments: 4

let f(x) =∫_0 ^1 lnt ln(1−xt)dt with ∣x∣<1 1)determine a explicit form for f(x) 2) find also g(x) =∫_0 ^1 ((tlnt)/(1−xt))dt 3) give f^((n)) (x) at form of integral 4) calculate ∫_0 ^1 ln(t)ln(1−t)dt and ∫_0 ^1 ln(t)ln(2−t)dt 5) calculate ∫_0 ^1 ((tln(t))/(2−t)) dt .

$${let}\:{f}\left({x}\right)\:=\int_{\mathrm{0}} ^{\mathrm{1}} {lnt}\:{ln}\left(\mathrm{1}−{xt}\right){dt}\:\:\:{with}\:\mid{x}\mid<\mathrm{1} \\ $$$$\left.\mathrm{1}\right){determine}\:{a}\:{explicit}\:{form}\:{for}\:{f}\left({x}\right) \\ $$$$\left.\mathrm{2}\right)\:{find}\:{also}\:{g}\left({x}\right)\:=\int_{\mathrm{0}} ^{\mathrm{1}} \:\frac{{tlnt}}{\mathrm{1}−{xt}}{dt} \\ $$$$\left.\mathrm{3}\right)\:{give}\:{f}^{\left({n}\right)} \left({x}\right)\:{at}\:{form}\:{of}\:{integral} \\ $$$$\left.\mathrm{4}\right)\:{calculate}\:\int_{\mathrm{0}} ^{\mathrm{1}} {ln}\left({t}\right){ln}\left(\mathrm{1}−{t}\right){dt}\:\:{and}\:\int_{\mathrm{0}} ^{\mathrm{1}} \:{ln}\left({t}\right){ln}\left(\mathrm{2}−{t}\right){dt} \\ $$$$\left.\mathrm{5}\right)\:{calculate}\:\int_{\mathrm{0}} ^{\mathrm{1}} \:\frac{{tln}\left({t}\right)}{\mathrm{2}−{t}}\:{dt}\:. \\ $$

Question Number 64662    Answers: 1   Comments: 3

∫(dx/((x^8 +x^4 +1)^2 )) ∫_(1/x) ^x ((ln(t))/(t^2 +1)) dt

$$\int\frac{{dx}}{\left({x}^{\mathrm{8}} +{x}^{\mathrm{4}} +\mathrm{1}\right)^{\mathrm{2}} } \\ $$$$ \\ $$$$\int_{\frac{\mathrm{1}}{{x}}} ^{{x}} \frac{{ln}\left({t}\right)}{{t}^{\mathrm{2}} +\mathrm{1}}\:{dt} \\ $$

Question Number 64649    Answers: 1   Comments: 1

calculate ∫_0 ^(2π) ((cosθ)/(5+3cosθ))dθ

$${calculate}\:\int_{\mathrm{0}} ^{\mathrm{2}\pi} \:\:\frac{{cos}\theta}{\mathrm{5}+\mathrm{3}{cos}\theta}{d}\theta \\ $$

Question Number 64642    Answers: 0   Comments: 1

∫(dx)/e^x +x

$$\int\left({dx}\right)/{e}^{{x}} +{x} \\ $$

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