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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-




Question Number 47060 by maxmathsup by imad last updated on 04/Nov/18
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) .
$${let}\:{f}\left({x}\right)\:=\:\int_{\mathrm{0}} ^{\mathrm{1}} \:\:\:\:\frac{{dt}}{\mathrm{2}+{ch}\left({xt}\right)} \\ $$$$\left.\mathrm{1}\right)\:{find}\:{a}\:{explicit}\:{form}\:{of}\:{f}\left({x}\right) \\ $$$$\left.\mathrm{2}\right)\:{calculate}\:{g}\left({x}\right)=\int_{\mathrm{0}} ^{\mathrm{1}} \:\:\frac{{tsh}\left({xt}\right)}{\left(\mathrm{2}+{ch}\left({xt}\right)\right)^{\mathrm{2}} }{dt} \\ $$$$\left.\mathrm{3}\right)\:{find}\:{the}\:{value}\:{of}\:\:\int_{\mathrm{0}} ^{\mathrm{1}} \:\:\:\frac{{dt}}{\mathrm{2}+{ch}\left(\mathrm{3}{t}\right)}\:{and}\:\:\int_{\mathrm{0}} ^{\mathrm{1}} \:\:\frac{{tsh}\left(\mathrm{3}{t}\right)}{\left(\mathrm{2}+{ch}\left(\mathrm{3}{t}\right)\right)^{\mathrm{2}} }{dt} \\ $$$$\left.\mathrm{4}\right)\:{calculate}\:{u}_{{n}} =\int_{\mathrm{0}} ^{\mathrm{1}} \:\:\:\:\frac{{dt}}{\mathrm{2}+{ch}\left({nt}\right)}\:{with}\:{n}\:{natural}\:{integr}\:\:{and}\:{study}\:{the}\:{convergence} \\ $$$${of}\:{the}\:{serie}\:\Sigma\:\frac{{u}_{{n}} }{{n}}\:. \\ $$

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