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IntegrationQuestion and Answers: Page 263

Question Number 47295 Answers: 0 Comments: 6

calculate f(α) =∫_(−∞) ^(+∞) (dx/(x^2 +2x cosα +1)) 2) calculate g(α)=∫_(−∞) ^(+∞) ((sinα)/((x^2 +2x cosα+1)^2 ))dx 3) find f^((n)) (α) with n integr natural . 4) calculate ∫_(−∞) ^(+∞) (dx/(x^2 +x +1)) and ∫_(−∞) ^(+∞) (dx/((x^2 +x+1)^2 ))

$${calculate}\:{f}\left(\alpha\right)\:=\int_{−\infty} ^{+\infty} \:\:\:\:\frac{{dx}}{{x}^{\mathrm{2}} \:+\mathrm{2}{x}\:{cos}\alpha\:+\mathrm{1}} \\ $$$$\left.\mathrm{2}\right)\:{calculate}\:{g}\left(\alpha\right)=\int_{−\infty} ^{+\infty} \:\:\frac{{sin}\alpha}{\left({x}^{\mathrm{2}} \:+\mathrm{2}{x}\:{cos}\alpha+\mathrm{1}\right)^{\mathrm{2}} }{dx} \\ $$$$\left.\mathrm{3}\right)\:{find}\:{f}^{\left({n}\right)} \left(\alpha\right)\:{with}\:{n}\:{integr}\:{natural}\:. \\ $$$$\left.\mathrm{4}\right)\:{calculate}\:\int_{−\infty} ^{+\infty} \:\:\:\frac{{dx}}{{x}^{\mathrm{2}} \:+{x}\:+\mathrm{1}}\:{and}\:\:\int_{−\infty} ^{+\infty} \:\:\:\:\frac{{dx}}{\left({x}^{\mathrm{2}} \:+{x}+\mathrm{1}\right)^{\mathrm{2}} } \\ $$

Question Number 47311 Answers: 0 Comments: 1

Question Number 47259 Answers: 0 Comments: 0

∫_0 ^1 ((tan^(−1) ((x/(x+1))))/(tan^(−1) (((1−2x^2 +2x)/2))))dx

$$\int_{\mathrm{0}} ^{\mathrm{1}} \frac{\boldsymbol{\mathrm{tan}}^{−\mathrm{1}} \left(\frac{\boldsymbol{\mathrm{x}}}{\boldsymbol{\mathrm{x}}+\mathrm{1}}\right)}{\boldsymbol{\mathrm{tan}}^{−\mathrm{1}} \left(\frac{\mathrm{1}−\mathrm{2}\boldsymbol{\mathrm{x}}^{\mathrm{2}} +\mathrm{2}\boldsymbol{\mathrm{x}}}{\mathrm{2}}\right)}\boldsymbol{\mathrm{dx}} \\ $$

Question Number 47248 Answers: 0 Comments: 1

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

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

Question Number 47120 Answers: 0 Comments: 0

calculate ∫_0 ^∞ e^(−t) ln(1+2t^2 )dt

$${calculate}\:\int_{\mathrm{0}} ^{\infty} {e}^{−{t}} {ln}\left(\mathrm{1}+\mathrm{2}{t}^{\mathrm{2}} \right){dt} \\ $$

Question Number 47119 Answers: 0 Comments: 1

let u_n =∫_(−∞) ^∞ e^(−nx^2 +x) dx 1)calculate u_n 2)find Σ_n u_n

$${let}\:{u}_{{n}} =\int_{−\infty} ^{\infty} \:{e}^{−{nx}^{\mathrm{2}} +{x}} {dx} \\ $$$$\left.\mathrm{1}\right){calculate}\:{u}_{{n}} \\ $$$$\left.\mathrm{2}\right){find}\:\sum_{{n}} \:{u}_{{n}} \\ $$

Question Number 47114 Answers: 0 Comments: 1

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

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

Question Number 47112 Answers: 1 Comments: 3

calculate ∫_0 ^(π/2) ln(cosx+sinx)ex

$${calculate}\:\:\:\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} {ln}\left({cosx}+{sinx}\right){ex} \\ $$

Question Number 47101 Answers: 1 Comments: 1

Question Number 47088 Answers: 1 Comments: 0

Question Number 47065 Answers: 0 Comments: 0

let v_n (a)= ∫_(1/n) ^n (1−(a/x^2 ))arctan(1+(a/x))dx with a>0 1) determine a explicit form of v_n (a) 2) study the convergence of Σ_n v_n (a) 3)calculate v_n (1) and Σ_n v_n (1) .

$${let}\:{v}_{{n}} \left({a}\right)=\:\int_{\frac{\mathrm{1}}{{n}}} ^{{n}} \:\:\left(\mathrm{1}−\frac{{a}}{{x}^{\mathrm{2}} }\right){arctan}\left(\mathrm{1}+\frac{{a}}{{x}}\right){dx}\:\:{with}\:{a}>\mathrm{0} \\ $$$$\left.\mathrm{1}\right)\:{determine}\:{a}\:{explicit}\:{form}\:{of}\:{v}_{{n}} \left({a}\right) \\ $$$$\left.\mathrm{2}\right)\:{study}\:{the}\:{convergence}\:{of}\:\sum_{{n}} \:{v}_{{n}} \left({a}\right) \\ $$$$\left.\mathrm{3}\right){calculate}\:{v}_{{n}} \left(\mathrm{1}\right)\:\:{and}\:\sum_{{n}} {v}_{{n}} \left(\mathrm{1}\right)\:. \\ $$

Question Number 47062 Answers: 1 Comments: 0

find ∫ (√(((√x)−1)/((√x)+1)))dx

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

Question Number 47061 Answers: 0 Comments: 0

let f(a) =∫ (√(1+atan(x)))dx 1) find a explicit form of f(a) 2) calculate ∫ (√(1+2tan(x)))dx .

$${let}\:\:{f}\left({a}\right)\:=\int\:\:\:\sqrt{\mathrm{1}+{atan}\left({x}\right)}{dx} \\ $$$$\left.\mathrm{1}\right)\:{find}\:{a}\:{explicit}\:{form}\:{of}\:{f}\left({a}\right) \\ $$$$\left.\mathrm{2}\right)\:{calculate}\:\int\:\:\sqrt{\mathrm{1}+\mathrm{2}{tan}\left({x}\right)}{dx}\:. \\ $$

Question Number 47060 Answers: 0 Comments: 0

let f(x) = ∫_0 ^1 (dt/(2+ch(xt))) 1) find a explicit form of f(x) 2) calculate g(x)=∫_0 ^1 ((tsh(xt))/((2+ch(xt))^2 ))dt 3) find the value of ∫_0 ^1 (dt/(2+ch(3t))) and ∫_0 ^1 ((tsh(3t))/((2+ch(3t))^2 ))dt 4) calculate u_n =∫_0 ^1 (dt/(2+ch(nt))) with n natural integr and study the convergence of the serie Σ (u_n /n) .

$${let}\:{f}\left({x}\right)\:=\:\int_{\mathrm{0}} ^{\mathrm{1}} \:\:\:\:\frac{{dt}}{\mathrm{2}+{ch}\left({xt}\right)} \\ $$$$\left.\mathrm{1}\right)\:{find}\:{a}\:{explicit}\:{form}\:{of}\:{f}\left({x}\right) \\ $$$$\left.\mathrm{2}\right)\:{calculate}\:{g}\left({x}\right)=\int_{\mathrm{0}} ^{\mathrm{1}} \:\:\frac{{tsh}\left({xt}\right)}{\left(\mathrm{2}+{ch}\left({xt}\right)\right)^{\mathrm{2}} }{dt} \\ $$$$\left.\mathrm{3}\right)\:{find}\:{the}\:{value}\:{of}\:\:\int_{\mathrm{0}} ^{\mathrm{1}} \:\:\:\frac{{dt}}{\mathrm{2}+{ch}\left(\mathrm{3}{t}\right)}\:{and}\:\:\int_{\mathrm{0}} ^{\mathrm{1}} \:\:\frac{{tsh}\left(\mathrm{3}{t}\right)}{\left(\mathrm{2}+{ch}\left(\mathrm{3}{t}\right)\right)^{\mathrm{2}} }{dt} \\ $$$$\left.\mathrm{4}\right)\:{calculate}\:{u}_{{n}} =\int_{\mathrm{0}} ^{\mathrm{1}} \:\:\:\:\frac{{dt}}{\mathrm{2}+{ch}\left({nt}\right)}\:{with}\:{n}\:{natural}\:{integr}\:\:{and}\:{study}\:{the}\:{convergence} \\ $$$${of}\:{the}\:{serie}\:\Sigma\:\frac{{u}_{{n}} }{{n}}\:. \\ $$

Question Number 47059 Answers: 2 Comments: 1

find ∫ (dx/(1+cos x +cos(2x)))

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

Question Number 47058 Answers: 1 Comments: 0

calculate ∫_0 ^1 ((1+x^3 )/(1+x^4 ))dx

$${calculate}\:\int_{\mathrm{0}} ^{\mathrm{1}} \:\:\frac{\mathrm{1}+{x}^{\mathrm{3}} }{\mathrm{1}+{x}^{\mathrm{4}} }{dx} \\ $$

Question Number 47026 Answers: 4 Comments: 4

Question Number 47018 Answers: 1 Comments: 1

find ∫ (√(x+2−(√(x−1))))dx

$${find}\:\int\:\sqrt{{x}+\mathrm{2}−\sqrt{{x}−\mathrm{1}}}{dx} \\ $$

Question Number 47098 Answers: 1 Comments: 8

prove that:lim_(n→∞) ∫_(−1) ^1 (1+(t/n))^n dt = e−(1/e).

$${prove}\:{that}:\underset{{n}\rightarrow\infty} {\mathrm{lim}}\:\int_{−\mathrm{1}} ^{\mathrm{1}} \left(\mathrm{1}+\frac{{t}}{{n}}\right)^{{n}} {dt}\:=\:{e}−\frac{\mathrm{1}}{{e}}. \\ $$

Question Number 46898 Answers: 2 Comments: 2

∫((tanx)/((tanx+1)^2 −2tan^2 x ))dx=??

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

Question Number 46856 Answers: 0 Comments: 1

find f(t) =∫_0 ^1 x^2 arctan(1+tx)dx

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

Question Number 46855 Answers: 0 Comments: 1

calculate ∫_0 ^1 x arctan(1+x)dx

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

Question Number 46854 Answers: 1 Comments: 1

find =∫_0 ^π ((sinx)/(2+cos(2x)))dx

$${find}\:\:=\int_{\mathrm{0}} ^{\pi} \:\:\:\frac{{sinx}}{\mathrm{2}+{cos}\left(\mathrm{2}{x}\right)}{dx} \\ $$

Question Number 46853 Answers: 1 Comments: 1

fnd ∫ (dx/(1+cos(tx)))

$${fnd}\:\:\int\:\:\:\:\:\:\frac{{dx}}{\mathrm{1}+{cos}\left({tx}\right)} \\ $$

Question Number 46851 Answers: 0 Comments: 3

let f(x)=∫_0 ^(2π) ((sint)/(x +sint))dt withx>1 1) calculate f(x) 2) calculate ∫_0 ^(2π) ((sint)/((x+sint)^2 ))dt 3)find the value of ∫_0 ^(2π) ((sint)/(2+sint))dt and ∫_0 ^(2π) ((sint)/((2+sint)^2 ))dt

$${let}\:{f}\left({x}\right)=\int_{\mathrm{0}} ^{\mathrm{2}\pi} \:\:\frac{{sint}}{{x}\:+{sint}}{dt}\:\:{withx}>\mathrm{1} \\ $$$$\left.\mathrm{1}\right)\:{calculate}\:{f}\left({x}\right) \\ $$$$\left.\mathrm{2}\right)\:{calculate}\:\int_{\mathrm{0}} ^{\mathrm{2}\pi} \:\:\:\frac{{sint}}{\left({x}+{sint}\right)^{\mathrm{2}} }{dt} \\ $$$$\left.\mathrm{3}\right){find}\:{the}\:{value}\:{of}\:\int_{\mathrm{0}} ^{\mathrm{2}\pi} \:\:\frac{{sint}}{\mathrm{2}+{sint}}{dt}\:{and}\:\int_{\mathrm{0}} ^{\mathrm{2}\pi} \:\:\frac{{sint}}{\left(\mathrm{2}+{sint}\right)^{\mathrm{2}} }{dt} \\ $$

Question Number 46850 Answers: 1 Comments: 1

let a^2 >b^(2 ) +c^2 calculate ∫_0 ^(2π) (dθ/(a+bsinθ +c cosθ))

$${let}\:{a}^{\mathrm{2}} >{b}^{\mathrm{2}\:} +{c}^{\mathrm{2}} \:\:{calculate}\:\int_{\mathrm{0}} ^{\mathrm{2}\pi} \:\:\frac{{d}\theta}{{a}+{bsin}\theta\:+{c}\:{cos}\theta} \\ $$

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