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IntegrationQuestion and Answers: Page 290
Question Number 33983 Answers: 0 Comments: 1
$${find}\:\int_{\mathrm{1}\left(\right.} ^{\infty} \frac{\mathrm{1}}{{x}}{ln}\left(\frac{{x}+\mathrm{1}}{{x}−\mathrm{1}}\right){dx}. \\ $$
Question Number 33980 Answers: 0 Comments: 0
$$\left.\mathrm{1}\right)\:{let}\:{consider}\:{f}\left({x}\right)=\mid{cosx}\mid\:\pi\:{periodix} \\ $$$${developp}\:{f}\:{at}\:{fourier}\:{serie} \\ $$$$\left.\mathrm{2}\right){find}\:{the}\:{valueof}\:\sum_{{n}=\mathrm{1}} ^{\infty} \:\:\frac{\left(−\mathrm{1}\right)^{{n}} }{\mathrm{4}{n}^{\mathrm{2}} \:−\mathrm{1}} \\ $$$$\left.\mathrm{3}\right){find}\:{the}\:{value}\:{of}\:\:\sum_{{n}=\mathrm{1}} ^{\infty} \:\:\:\frac{\mathrm{1}}{\left(\mathrm{4}{n}^{\mathrm{2}} −\mathrm{1}\right)^{\mathrm{2}} }\:. \\ $$
Question Number 33979 Answers: 0 Comments: 1
$${we}\:{give}\:{for}\:{t}>\mathrm{0}\:\int_{\mathrm{0}} ^{\infty} \:\:\frac{{sinx}}{{x}}\:{e}^{−{tx}} {dx}\:=\frac{\pi}{\mathrm{2}}\:−{arctant} \\ $$$${use}\:{this}\:{result}\:{to}\:{find}\:{the}\:{value}\:{of}\:\int_{\mathrm{0}} ^{\infty} \:\:\frac{\left(\mathrm{1}−{e}^{−{x}} \right){sinx}}{{x}^{\mathrm{2}} }{dx}\:. \\ $$
Question Number 33978 Answers: 1 Comments: 2
$${let}\:{f}\left({t}\right)\:=\:\int_{\mathrm{0}} ^{\infty} \:\:\:\frac{{sin}\left({x}^{\mathrm{2}} \right){e}^{−{tx}^{\mathrm{2}} } }{{x}^{\mathrm{2}} }\:{dx}\:\:\:\:\:\:\:{with}\:{t}>\mathrm{0} \\ $$$${find}\:{a}\:{simple}\:{form}\:{of}\:{f}^{'} \left({t}\right)\:. \\ $$
Question Number 33915 Answers: 2 Comments: 3
$${let}\:\Gamma\left({x}\right)=\int_{\mathrm{0}} ^{\infty} \:{t}^{{x}−\mathrm{1}} \:{e}^{−{t}} \:{dt}\:{with}\:{x}>\mathrm{0} \\ $$$$\left.\mathrm{1}\right)\:{find}\:\Gamma^{\left({n}\right)} \left({x}\right)\:{with}\:{n}\in\:{N}^{\bigstar} \\ $$$$\left.\mathrm{2}\right)\:{calculate}\:\Gamma\left({n}\:+\frac{\mathrm{3}}{\mathrm{2}}\right)\:{for}\:{n}\:{integr}. \\ $$
Question Number 33896 Answers: 3 Comments: 0
$${let}\:\Gamma\left({x}\right)=\int_{\mathrm{0}} ^{\infty} \:\:{t}^{{x}−\mathrm{1}} \:{e}^{−{t}} {dt} \\ $$$$\left.\mathrm{1}\right)\:{find}\:\Gamma\left({x}+\mathrm{1}\right)\:{interms}\:{of}\:\Gamma\left({x}\right)\:\:\:{with}\:{x}>\mathrm{0} \\ $$$$\left.\mathrm{2}\right){calculate}\:\Gamma\left({n}\right)\:{for}\:{n}\:\in\:{N}^{\bigstar} \\ $$$$\left.\mathrm{3}\right){calculate}\:\Gamma\left(\frac{\mathrm{3}}{\mathrm{2}}\right)\:. \\ $$
Question Number 33895 Answers: 3 Comments: 0
$${let}\:\Gamma\left({x}\right)=\int_{\mathrm{0}} ^{\infty} \:{t}^{{x}−\mathrm{1}} \:{e}^{−{t}} {dt}\:\:{with}\:{x}>\mathrm{0} \\ $$$$\left.\mathrm{1}\right)\:{prove}\:{that}\:\Gamma\left({x}\right)\Gamma\left(\mathrm{1}−{x}\right)=\:\frac{\pi}{{sin}\left(\pi{x}\right)} \\ $$$$\left.\mathrm{2}\right)\:{find}\:{the}\:{value}\:{of}\:\:\int_{\mathrm{0}} ^{\infty} \:{e}^{−{x}^{\mathrm{2}} } {dx}\:. \\ $$
Question Number 33894 Answers: 0 Comments: 1
$$\left.\mathrm{1}\right){let}\:{f}\:\:{R}\rightarrow{C}\:\:\mathrm{2}\pi\:{periodic}\:{even}\:\:/{f}\left({x}\right)={x}\: \\ $$$$\forall\:{x}\in\left[\mathrm{0},\pi\left[\:\:{developp}\:{f}\:{at}\:{fourier}\:{serie}\right.\right. \\ $$$$\left.\mathrm{2}\right)\:{calculate}\:\:\sum_{{p}=\mathrm{0}} ^{\infty} \:\:\frac{\mathrm{1}}{\left(\mathrm{2}{p}+\mathrm{1}\right)^{\mathrm{2}} }\:. \\ $$
Question Number 33888 Answers: 0 Comments: 0
$${developp}\:{at}\:{integr}\:{serie}\:{f}\left({x}\right)=\:\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \:\:\:\frac{{dt}}{\sqrt{\mathrm{1}−{x}^{\mathrm{2}} {sin}^{\mathrm{2}} {t}}}\:. \\ $$$${with}\:\mid{x}\mid<\mathrm{1}\:. \\ $$
Question Number 33885 Answers: 0 Comments: 1
$${developp}\:{at}\:{integr}\:{serie}\:{f}\left({x}\right)=\:\int_{\mathrm{0}} ^{{x}} {sin}\left({t}^{\mathrm{2}} \right){dt}\:. \\ $$
Question Number 33884 Answers: 0 Comments: 1
$${let}\:{F}\left({x}\right)=\:\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \:\frac{{arctan}\left({xtant}\right)}{{tant}}\:{dt}\:{find}\:{a}\:{simple} \\ $$$${form}\:{of}\:{f}\left({x}\right)\:. \\ $$$$\left.\mathrm{2}\right)\:{find}\:{the}\:{value}\:{of}\:\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \:\:\frac{{arctan}\left(\mathrm{2}{tant}\right)}{{tant}}{dt}\:. \\ $$
Question Number 33883 Answers: 0 Comments: 1
$${find}\:{a}\:{simple}\:{form}\:{of}\:{f}\left({x}\right)=\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} {ln}\left(\mathrm{1}+{xsin}^{\mathrm{2}} {t}\right){dt} \\ $$$${with}\:\mid{x}\mid<\mathrm{1}. \\ $$
Question Number 33845 Answers: 0 Comments: 1
$${let}\:{I}_{{n}} =\:\int_{\mathrm{0}} ^{\mathrm{1}} \:\:\:\frac{{arctan}\left(\mathrm{1}\:+{n}\right)}{\sqrt{\mathrm{1}+{x}^{{n}} }}\:{find}\:{lim}_{{n}\rightarrow+\infty} \:{I}_{{n}} \:. \\ $$
Question Number 33835 Answers: 0 Comments: 1
$${find}\:{the}\:{value}\:{of}\:\:\:\int_{−\infty} ^{+\infty} \:\:\:\:\frac{{cos}\left(\pi{x}\right)}{\left({x}^{\mathrm{2}} \:+\mathrm{1}+{i}\right)^{\mathrm{2}} }\:{dx} \\ $$
Question Number 33787 Answers: 0 Comments: 0
$$\underset{{n}\rightarrow\infty} {\mathrm{lim}}\:\left(\frac{\mathrm{1}}{{n}}\:\underset{\mathrm{1}} {\overset{{n}} {\int}}\:{n}^{\frac{\mathrm{1}}{{x}}} \:{dx}\right) \\ $$
Question Number 33823 Answers: 1 Comments: 0
$$\:\:{solve}\::\: \\ $$$$\:{I}\:=\:\underset{\mathrm{0}} {\overset{\pi} {\int}}\:\frac{\left({r}−{R}\:{cos}\theta\right)\:{sin}\:\theta\:}{\left({R}^{\mathrm{2}\:} +\:{r}^{\mathrm{2}} \:−\:\mathrm{2}{Rr}\:{cos}\:\theta\right)^{\mathrm{3}/\mathrm{2}} }\:{d}\theta \\ $$$${for}\:\:\:{r}\:<\:{R} \\ $$$${and}\:{r}\:>\:{R}\:\:{respectively}. \\ $$
Question Number 33759 Answers: 0 Comments: 9
$${solve}\::\: \\ $$$$\:\underset{−\pi/\mathrm{2}} {\overset{\pi/\mathrm{2}} {\int}}\:\:\frac{{sin}\:\theta\:}{\sqrt{\:{R}^{\mathrm{2}} \:+\:{r}^{\mathrm{2}} \:−\:\mathrm{2}{rR}\:{cos}\:\theta}}\:{d}\theta \\ $$
Question Number 33747 Answers: 0 Comments: 0
$${Calculate}\:\int_{−\infty} ^{+\infty} {e}^{−{x}^{\mathrm{2}} } {dx}\:\:{using}\:\:{Residue}\:{theorem} \\ $$
Question Number 33744 Answers: 0 Comments: 1
$${let}\:\:{P}_{{n}} \left({x}\right)=\left(\mathrm{1}+{x}^{\mathrm{2}} \right)\left(\mathrm{1}+{x}^{\mathrm{4}} \right)....\left(\mathrm{1}+{x}^{\mathrm{2}^{{n}} } \right) \\ $$$${calculate}\:\:{lim}_{{n}\rightarrow+\infty} \int_{\mathrm{0}} ^{{x}} \:{P}_{{n}} \left({t}\right){dt}\:\:{with}\:\:\mathrm{0}<{x}<\mathrm{1}\:. \\ $$
Question Number 33737 Answers: 1 Comments: 3
$${find}\:{the}\:{value}\:{of}\:\:\int_{\mathrm{0}} ^{\infty} \:\:\:\:\frac{{cos}\left({xt}\right)}{\left({t}^{\mathrm{2}} \:+\:{x}^{\mathrm{2}} \right)^{\mathrm{2}} }\:{dt}\:. \\ $$
Question Number 33736 Answers: 2 Comments: 1
$${find}\:{the}\:{value}\:{of}\:\:\int_{−\infty} ^{+\infty} \:\:\:\:\:\frac{{x}^{\mathrm{2}} }{\left(\mathrm{1}+{x}\:+{x}^{\mathrm{2}} \right)^{\mathrm{2}} }{dx} \\ $$
Question Number 33735 Answers: 0 Comments: 1
$${calculate}\:\:\int_{\mathrm{0}} ^{\infty} \:\:\:\:\:\:\frac{{cos}\left(\mathrm{2}{x}\right){dx}}{\left({x}^{\mathrm{2}} +\mathrm{1}\right)\left(\:\mathrm{2}{x}^{\mathrm{2}} \:+\mathrm{3}\right)}\:. \\ $$
Question Number 33705 Answers: 1 Comments: 1
$${let}\:\:\alpha>\mathrm{0}\:\:{find}\:{the}\:{fourier}\:{transform}\:{of} \\ $$$${f}\left({t}\right)\:=\:{e}^{−{a}^{\mathrm{2}} {t}^{\mathrm{2}} } \\ $$
Question Number 33704 Answers: 0 Comments: 1
$${let}\:{f}\left({t}\right)\:=\:\frac{\mathrm{1}}{{a}^{\mathrm{2}} \:+{t}^{\mathrm{2}} }\:\:{witha}>\mathrm{0}\:{give}\:{the}\:{fourier} \\ $$$${transformfor}\:{f}\:. \\ $$$$ \\ $$
Question Number 33703 Answers: 0 Comments: 0
$${give}\:\int_{\mathrm{0}} ^{\infty} \:\:\:\:\:\frac{{x}\:{e}^{−{x}} }{\mathrm{1}\:−{e}^{−\mathrm{2}{x}} }\:{sin}\left(\pi{x}\right){dx}\:\:{at}\:{form}\:{of}\:{serie}. \\ $$
Question Number 33695 Answers: 0 Comments: 1
$${find}\:{lim}_{{n}\rightarrow+\infty} \:\int_{\mathrm{0}} ^{\infty} \:\:\:\:\:\frac{{e}^{−\frac{{x}}{{n}}} }{\mathrm{1}+{x}^{\mathrm{2}} }{dx}. \\ $$
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