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IntegrationQuestion and Answers: Page 263

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))

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

Question Number 46849 Answers: 0 Comments: 0

let A_p =Σ_(n=1) ^∞ n^p x^n with p integr . and x ∈]−1,1[ . 1) calculate A_1 ,A_2 and A_3 2) find a relation of recurrence betwen the A_n 3) calculate Σ_(n=1) ^∞ n^4 x^n and Σ_(n=1) ^∞ n^5 x^n .

$$\left.{let}\:{A}_{{p}} =\sum_{{n}=\mathrm{1}} ^{\infty} \:{n}^{{p}} {x}^{{n}} \:\:\:\:{with}\:{p}\:{integr}\:.\:{and}\:{x}\:\in\right]−\mathrm{1},\mathrm{1}\left[\:.\right. \\ $$$$\left.\mathrm{1}\right)\:{calculate}\:{A}_{\mathrm{1}} ,{A}_{\mathrm{2}} \:{and}\:{A}_{\mathrm{3}} \\ $$$$\left.\mathrm{2}\right)\:{find}\:{a}\:{relation}\:{of}\:{recurrence}\:\:{betwen}\:{the}\:{A}_{{n}} \\ $$$$\left.\mathrm{3}\right)\:{calculate}\:\sum_{{n}=\mathrm{1}} ^{\infty} \:{n}^{\mathrm{4}} {x}^{{n}} \:{and}\:\sum_{{n}=\mathrm{1}} ^{\infty} \:{n}^{\mathrm{5}} {x}^{{n}} \:. \\ $$

Question Number 46848 Answers: 0 Comments: 1

caculate ∫∫_D (x^2 −y^2 ) e^(−x^2 −y^2 ) dxdy with D ={(x,y)∈R^2 / x^2 +y^2 ≤4}

$${caculate}\:\:\int\int_{{D}} \:\:\left({x}^{\mathrm{2}} −{y}^{\mathrm{2}} \right)\:{e}^{−{x}^{\mathrm{2}} \:−{y}^{\mathrm{2}} } {dxdy}\:\:{with} \\ $$$${D}\:=\left\{\left({x},{y}\right)\in{R}^{\mathrm{2}} /\:\:\:{x}^{\mathrm{2}} \:+{y}^{\mathrm{2}} \:\leqslant\mathrm{4}\right\} \\ $$

Question Number 46847 Answers: 0 Comments: 1

calculate ∫∫_(0≤x≤1 and 1≤y≤2) e^(x/y) dxdy

$${calculate}\:\:\int\int_{\mathrm{0}\leqslant{x}\leqslant\mathrm{1}\:{and}\:\mathrm{1}\leqslant{y}\leqslant\mathrm{2}} \:\:{e}^{\frac{{x}}{{y}}} {dxdy} \\ $$

Question Number 46846 Answers: 0 Comments: 1

calculate ∫∫_D ((x+y)/(√(1−x^2 −y^2 )))dxdy with D={(x,y)∈R^2 /x≥0,y≥0,x^2 +y^2 <1}

$${calculate}\:\int\int_{{D}} \:\:\:\:\frac{{x}+{y}}{\sqrt{\mathrm{1}−{x}^{\mathrm{2}} −{y}^{\mathrm{2}} }}{dxdy}\:{with}\:{D}=\left\{\left({x},{y}\right)\in{R}^{\mathrm{2}} /{x}\geqslant\mathrm{0},{y}\geqslant\mathrm{0},{x}^{\mathrm{2}} \:+{y}^{\mathrm{2}} <\mathrm{1}\right\} \\ $$

Question Number 46845 Answers: 0 Comments: 0

calculate ∫_0 ^1 (e^(−x) /(1+x)) dx .