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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 47064 Answers: 0 Comments: 1

1)calculate u_n =∫_0 ^∞ ((sin(nx))/(sh(2x)))dx with n integr natural 2) calculate Σ_(n=0) ^∞ u_n .

$$\left.\mathrm{1}\right){calculate}\:\:{u}_{{n}} =\int_{\mathrm{0}} ^{\infty} \:\:\frac{{sin}\left({nx}\right)}{{sh}\left(\mathrm{2}{x}\right)}{dx}\:\:{with}\:\:{n}\:{integr}\:{natural} \\ $$$$\left.\mathrm{2}\right)\:{calculate}\:\sum_{{n}=\mathrm{0}} ^{\infty} \:{u}_{{n}} \:. \\ $$

Question Number 47063 Answers: 0 Comments: 0

1)calculate u_n = ∫_0 ^∞ e^(−nx) ln(1+x)dx with n integr natural 2) calculate lim_(n→+∞) u_n 3) find Σ_(n=0) ^∞ u_n

$$\left.\mathrm{1}\right){calculate}\:{u}_{{n}} =\:\int_{\mathrm{0}} ^{\infty} \:\:{e}^{−{nx}} {ln}\left(\mathrm{1}+{x}\right){dx}\:\:{with}\:{n}\:{integr}\:{natural} \\ $$$$\left.\mathrm{2}\right)\:{calculate}\:{lim}_{{n}\rightarrow+\infty} {u}_{{n}} \\ $$$$\left.\mathrm{3}\right)\:{find}\:\sum_{{n}=\mathrm{0}} ^{\infty} \:{u}_{{n}} \\ $$

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 47043 Answers: 0 Comments: 0

if p is a prime number and (a,p)= then prove that a^(p−1) ≡ 1(mod p)

$${if}\:{p}\:{is}\:{a}\:{prime}\:{number}\:{and}\:\left({a},{p}\right)=\:{then}\:{prove}\:{that}\:{a}^{{p}−\mathrm{1}} \equiv\:\mathrm{1}\left({mod}\:{p}\right) \\ $$

Question Number 47036 Answers: 0 Comments: 0

let m,n denotes any two possitive,relative prime integers, then prove that φ(mn)=φ(m)∙φ(n)