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IntegrationQuestion and Answers: Page 217

Question Number 65319 Answers: 3 Comments: 2

Question Number 65293 Answers: 0 Comments: 3

1) calculate ∫_(−∞) ^(+∞) (dx/(x−a)) with a ∈C 2) find the values of ∫_0 ^∞ (dx/(x^4 +1)) and ∫_0 ^∞ (dx/(x^6 +1)) by using the decomposition inside C(x).

$$\left.\mathrm{1}\right)\:{calculate}\:\int_{−\infty} ^{+\infty} \:\:\frac{{dx}}{{x}−{a}}\:\:{with}\:{a}\:\in{C} \\ $$$$\left.\mathrm{2}\right)\:{find}\:{the}\:{values}\:{of}\:\:\int_{\mathrm{0}} ^{\infty} \:\:\:\:\frac{{dx}}{{x}^{\mathrm{4}} \:+\mathrm{1}}\:\:{and}\:\:\int_{\mathrm{0}} ^{\infty} \:\:\frac{{dx}}{{x}^{\mathrm{6}} \:+\mathrm{1}} \\ $$$${by}\:{using}\:{the}\:{decomposition}\:{inside}\:{C}\left({x}\right). \\ $$

Question Number 65290 Answers: 1 Comments: 3

f(x) =∫_0 ^1 (dt/(x+e^t )) with 0≤x≤1 1) find aexplicit form of f(x) 2) calculate ∫_0 ^1 (dt/(2+e^t )) 3) find g(x) =∫_0 ^1 (dt/((x+e^t )^2 )) 4) calculate ∫_0 ^1 (dt/((1+e^t )^2 )) 5) give f^((n)) (x) at form of integrals 6) developp f at integr serie.

$${f}\left({x}\right)\:=\int_{\mathrm{0}} ^{\mathrm{1}} \:\:\frac{{dt}}{{x}+{e}^{{t}} }\:\:\:{with}\:\:\mathrm{0}\leqslant{x}\leqslant\mathrm{1} \\ $$$$\left.\mathrm{1}\right)\:{find}\:{aexplicit}\:{form}\:{of}\:{f}\left({x}\right) \\ $$$$\left.\mathrm{2}\right)\:{calculate}\:\int_{\mathrm{0}} ^{\mathrm{1}} \:\:\frac{{dt}}{\mathrm{2}+{e}^{{t}} } \\ $$$$\left.\mathrm{3}\right)\:{find}\:{g}\left({x}\right)\:=\int_{\mathrm{0}} ^{\mathrm{1}} \:\frac{{dt}}{\left({x}+{e}^{{t}} \right)^{\mathrm{2}} } \\ $$$$\left.\mathrm{4}\right)\:{calculate}\:\int_{\mathrm{0}} ^{\mathrm{1}} \:\:\frac{{dt}}{\left(\mathrm{1}+{e}^{{t}} \right)^{\mathrm{2}} } \\ $$$$\left.\mathrm{5}\right)\:{give}\:{f}^{\left({n}\right)} \left({x}\right)\:{at}\:{form}\:{of}\:{integrals} \\ $$$$\left.\mathrm{6}\right)\:{developp}\:{f}\:{at}\:{integr}\:{serie}. \\ $$

Question Number 65288 Answers: 0 Comments: 4

1) let f(x) =∫_0 ^(+∞) (dt/(t^3 +x^3 )) with x>0 calculate f(x) 2) find also g(x) =∫_0 ^∞ (dt/((t^3 +x^3 )^2 )) 3) find the values of integrals ∫_0 ^∞ (dt/(t^3 +1)) and ∫_0 ^∞ (dt/((t^3 +1)^2 )) 4) give f^((n)) (x) at form of integrals.

$$\left.\mathrm{1}\right)\:{let}\:{f}\left({x}\right)\:=\int_{\mathrm{0}} ^{+\infty} \:\:\:\frac{{dt}}{{t}^{\mathrm{3}} \:+{x}^{\mathrm{3}} }\:\:\:{with}\:{x}>\mathrm{0} \\ $$$${calculate}\:{f}\left({x}\right) \\ $$$$\left.\mathrm{2}\right)\:{find}\:{also}\:{g}\left({x}\right)\:=\int_{\mathrm{0}} ^{\infty} \:\:\frac{{dt}}{\left({t}^{\mathrm{3}} \:+{x}^{\mathrm{3}} \right)^{\mathrm{2}} } \\ $$$$\left.\mathrm{3}\right)\:{find}\:{the}\:{values}\:{of}\:{integrals}\:\:\int_{\mathrm{0}} ^{\infty} \:\:\:\frac{{dt}}{{t}^{\mathrm{3}} \:+\mathrm{1}}\:\:{and}\:\int_{\mathrm{0}} ^{\infty} \frac{{dt}}{\left({t}^{\mathrm{3}} \:+\mathrm{1}\right)^{\mathrm{2}} } \\ $$$$\left.\mathrm{4}\right)\:{give}\:{f}^{\left({n}\right)} \left({x}\right)\:{at}\:{form}\:{of}\:{integrals}. \\ $$

Question Number 65287 Answers: 0 Comments: 1

let f(x) =x∣x∣ 2π periodic odd developp f at fourier series

$${let}\:{f}\left({x}\right)\:={x}\mid{x}\mid\:\:\:\:\mathrm{2}\pi\:{periodic}\:\:{odd} \\ $$$${developp}\:{f}\:{at}\:{fourier}\:{series} \\ $$

Question Number 65286 Answers: 0 Comments: 2

1)find f(a)=∫_(−∞) ^(+∞) e^(−ax^2 ) cos(3−x^2 )dx with a>0 2) find the value of ∫_0 ^∞ e^(−3x^2 ) cos(3−x^2 )dx

$$\left.\mathrm{1}\right){find}\:\:\:{f}\left({a}\right)=\int_{−\infty} ^{+\infty} \:\:{e}^{−{ax}^{\mathrm{2}} } {cos}\left(\mathrm{3}−{x}^{\mathrm{2}} \right){dx}\:\:{with}\:{a}>\mathrm{0} \\ $$$$\left.\mathrm{2}\right)\:{find}\:{the}\:{value}\:{of}\:\int_{\mathrm{0}} ^{\infty} \:{e}^{−\mathrm{3}{x}^{\mathrm{2}} } {cos}\left(\mathrm{3}−{x}^{\mathrm{2}} \right){dx} \\ $$$$ \\ $$

Question Number 65203 Answers: 1 Comments: 1

∫_0 ^∞ (x^2 /(x^4 +x^2 +1))dx

$$\int_{\mathrm{0}} ^{\infty} \:\frac{{x}^{\mathrm{2}} }{{x}^{\mathrm{4}} +{x}^{\mathrm{2}} +\mathrm{1}}{dx} \\ $$

Question Number 65202 Answers: 1 Comments: 0

∫_0 ^π ((cos 2θ)/(1−2acos θ+a^2 ))dθ, a^2 <1 answer?

$$\int_{\mathrm{0}} ^{\pi} \:\frac{\mathrm{cos}\:\mathrm{2}\theta}{\mathrm{1}−\mathrm{2}{a}\mathrm{cos}\:\theta+{a}^{\mathrm{2}} }{d}\theta,\:{a}^{\mathrm{2}} <\mathrm{1} \\ $$$$\mathrm{answer}? \\ $$

Question Number 65200 Answers: 2 Comments: 0

Question Number 65196 Answers: 0 Comments: 0

find ∫_0 ^1 ((1/(1−x)) +(1/(lnx)))dx

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

Question Number 65195 Answers: 0 Comments: 0

find ∫_(π/4) ^(π/2) ln(ln(tanx)dx

$${find}\:\int_{\frac{\pi}{\mathrm{4}}} ^{\frac{\pi}{\mathrm{2}}} \:{ln}\left({ln}\left({tanx}\right){dx}\right. \\ $$

Question Number 65194 Answers: 0 Comments: 0

calculate ∫_0 ^1 ln(Γ(x))dx with Γ(x) =∫_0 ^∞ t^(x−1) e^(−t) dt and x>0

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

Question Number 65162 Answers: 0 Comments: 0

Question Number 65137 Answers: 0 Comments: 3

∫(dx/((x−2)^3 (x+1)^2 ))=?

$$\int\frac{{dx}}{\left({x}−\mathrm{2}\right)^{\mathrm{3}} \left({x}+\mathrm{1}\right)^{\mathrm{2}} }=? \\ $$

Question Number 65133 Answers: 0 Comments: 1

find f(x)=∫_0 ^(π/4) ln(sint +xcost)dt x real.

$${find}\:{f}\left({x}\right)=\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{4}}} {ln}\left({sint}\:+{xcost}\right){dt} \\ $$$${x}\:{real}. \\ $$

Question Number 65132 Answers: 0 Comments: 0

calculate ∫_0 ^∞ ((tarctan(2t))/(1+t^4 ))dt

$${calculate}\:\int_{\mathrm{0}} ^{\infty} \:\frac{{tarctan}\left(\mathrm{2}{t}\right)}{\mathrm{1}+{t}^{\mathrm{4}} }{dt} \\ $$

Question Number 65131 Answers: 0 Comments: 1

calculate ∫_0 ^∞ (dx/((x^4 −4i)^3 ))

$${calculate}\:\int_{\mathrm{0}} ^{\infty} \:\:\frac{{dx}}{\left({x}^{\mathrm{4}} −\mathrm{4}{i}\right)^{\mathrm{3}} } \\ $$

Question Number 65130 Answers: 0 Comments: 2

calculate ∫_0 ^∞ ((x^2 −3)/((x^4 +x^2 +2)^2 ))dx

$${calculate}\:\int_{\mathrm{0}} ^{\infty} \:\:\:\frac{{x}^{\mathrm{2}} −\mathrm{3}}{\left({x}^{\mathrm{4}} +{x}^{\mathrm{2}} +\mathrm{2}\right)^{\mathrm{2}} }{dx} \\ $$

Question Number 65129 Answers: 1 Comments: 1

calculate ∫_(−∞) ^(+∞) (dx/((x^2 +x+1)^3 ))

$${calculate}\:\int_{−\infty} ^{+\infty} \:\:\frac{{dx}}{\left({x}^{\mathrm{2}} \:+{x}+\mathrm{1}\right)^{\mathrm{3}} } \\ $$

Question Number 65100 Answers: 0 Comments: 8

∫_0 ^π (dθ/((a+cosθ)^2 )), a>1

$$\int_{\mathrm{0}} ^{\pi} \frac{{d}\theta}{\left({a}+{cos}\theta\right)^{\mathrm{2}} },\:{a}>\mathrm{1} \\ $$

Question Number 65092 Answers: 0 Comments: 2

calculate ∫ (1/(x cosx))Π_(i=1) ^n (1−tan^2 ((x/2^i )))dx

$${calculate}\:\:\int\:\:\frac{\mathrm{1}}{{x}\:{cosx}}\prod_{{i}=\mathrm{1}} ^{{n}} \left(\mathrm{1}−{tan}^{\mathrm{2}} \left(\frac{{x}}{\mathrm{2}^{{i}} }\right)\right){dx} \\ $$

Question Number 65061 Answers: 0 Comments: 2

let f(x) =∫_0 ^∞ (dt/((x−t +t^2 )^3 )) with x>(1/4) 1) calculate f(x) 2) calculate also g(x) =∫_0 ^∞ (dt/((x−t+t^2 )^4 )) 3)find the values of ∫_0 ^∞ (dt/((1−t+t^2 )^3 )) and ∫_0 ^∞ (dt/((2−t+t^2 )^4 ))

$${let}\:{f}\left({x}\right)\:=\int_{\mathrm{0}} ^{\infty} \:\:\:\frac{{dt}}{\left({x}−{t}\:+{t}^{\mathrm{2}} \right)^{\mathrm{3}} }\:\:{with}\:\:\:{x}>\frac{\mathrm{1}}{\mathrm{4}} \\ $$$$\left.\mathrm{1}\right)\:{calculate}\:{f}\left({x}\right) \\ $$$$\left.\mathrm{2}\right)\:{calculate}\:{also}\:\:{g}\left({x}\right)\:=\int_{\mathrm{0}} ^{\infty} \:\:\:\:\:\frac{{dt}}{\left({x}−{t}+{t}^{\mathrm{2}} \right)^{\mathrm{4}} } \\ $$$$\left.\mathrm{3}\right){find}\:{the}\:{values}\:{of}\:\int_{\mathrm{0}} ^{\infty} \:\:\:\frac{{dt}}{\left(\mathrm{1}−{t}+{t}^{\mathrm{2}} \right)^{\mathrm{3}} }\:\:{and}\:\int_{\mathrm{0}} ^{\infty} \:\:\frac{{dt}}{\left(\mathrm{2}−{t}+{t}^{\mathrm{2}} \right)^{\mathrm{4}} } \\ $$$$ \\ $$

Question Number 65059 Answers: 0 Comments: 1

calculate ∫_0 ^(+∞) (dx/((x^2 −x+1)^4 ))

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

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)

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

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