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IntegrationQuestion and Answers: Page 298
Question Number 32741 Answers: 0 Comments: 0
$${find}\:\int_{\mathrm{0}} ^{\mathrm{1}} \:\:\:\frac{{ln}\left({t}^{\mathrm{2}} \:+\mathrm{2}{t}\:{cosx}\:+\mathrm{1}\right)}{{t}}{dt}\:. \\ $$
Question Number 32740 Answers: 0 Comments: 2
$${find}\int_{\mathrm{0}} ^{\infty} \:\frac{{ln}\left({x}^{\mathrm{2}} \:+{t}^{\mathrm{2}} \right)}{\mathrm{1}+{t}^{\mathrm{2}} }{dt} \\ $$
Question Number 32739 Answers: 0 Comments: 1
$${let}\:{f}\left({x}\right)=\int_{\mathrm{0}} ^{\infty} \:\:\:\frac{{e}^{−{t}} }{\mathrm{1}+{xt}}{dt} \\ $$$${calculate}\:{f}^{\left({n}\right)} \left(\mathrm{0}\right). \\ $$
Question Number 32737 Answers: 1 Comments: 0
$${let}\:{give}\:\mathrm{0}\leqslant{x}\leqslant\mathrm{1}\:\:{calculate}\:\:\int_{\mathrm{0}} ^{\infty} \frac{{arctan}\left(\frac{{x}}{{t}}\right)}{\mathrm{1}+{t}^{\mathrm{2}} }\:{dt} \\ $$
Question Number 32736 Answers: 0 Comments: 0
$${let}\:{o}\leqslant{x}\leqslant\mathrm{1}\:\:{find}\:\int_{\mathrm{0}} ^{{x}} \:\frac{{lnt}}{{t}^{\mathrm{2}} −\mathrm{1}}{dt}\: \\ $$
Question Number 32733 Answers: 0 Comments: 0
$${prove}\:{that}\:\:\sum_{{n}=\mathrm{0}} ^{\infty} \:\:\frac{\mathrm{1}}{\left({n}!\right)^{\mathrm{2}} }\:=\frac{\mathrm{1}}{\mathrm{2}\pi}\:\int_{\mathrm{0}} ^{\mathrm{2}\pi} \:\:{e}^{\mathrm{2}{cosx}} {dx}\:. \\ $$
Question Number 32731 Answers: 0 Comments: 0
$$\left.\mathrm{1}\right)\:{prove}\:{that}\:\:\int_{\mathrm{0}} ^{\mathrm{1}} \:\frac{{arctant}}{{t}}{dt}=−\int_{\mathrm{0}} ^{\mathrm{1}} \:\:\frac{{lnt}}{\mathrm{1}+{t}^{\mathrm{2}} }{dt} \\ $$$$\left.\mathrm{2}\right)\:{find}\:\int_{\mathrm{0}} ^{\mathrm{1}} \:\frac{{arctant}}{{t}}{dt}\:{at}\:{form}\:{of}\:{serie} \\ $$
Question Number 32729 Answers: 0 Comments: 0
$${find}\:{lim}_{{n}\rightarrow\infty} \:\int_{\mathrm{0}} ^{{n}} \left({cos}\left(\frac{{x}}{{n}}\right)\right)^{{n}^{\mathrm{2}} } \:{dx}. \\ $$
Question Number 32724 Answers: 0 Comments: 0
$${let}\:{A}_{{n}} =\:\int_{\mathrm{0}} ^{{n}} \sqrt{\mathrm{1}+\left(\mathrm{1}−\frac{{x}}{{n}}\right)^{{n}} }\:{dt}. \\ $$$${find}\:{a}\:{rquivalent}\:{of}\:{A}_{{n}} . \\ $$
Question Number 32721 Answers: 0 Comments: 0
$${let}\:{x}>\mathrm{0}\:{and}\:{f}\left({x}\right)=\int_{{x}} ^{+\infty} \:\:\frac{{e}^{−{t}} }{{t}}{dt} \\ $$$$\left.\mathrm{1}\right){calculate}\:{f}^{'} \left({x}\right) \\ $$$$\left.\mathrm{2}\right)\:{find}\:{lim}_{{x}\rightarrow+\infty} {xf}\left({x}\right)\:{and}\:{lim}_{{x}\rightarrow\mathrm{0}^{+} } {xf}\left({x}\right). \\ $$
Question Number 32720 Answers: 0 Comments: 0
$${find}\:\int_{−\infty} ^{+\infty} \:\:\:\:\:\frac{{dt}}{\mathrm{1}\:+\left({t}+\mathrm{2}{i}\right)^{\mathrm{2}} }\:. \\ $$
Question Number 32719 Answers: 0 Comments: 0
$${cslculate}\:\:\int_{\mathrm{0}} ^{\infty} \:\left({t}\:−\left[{t}\right]\right){e}^{−\mathrm{3}{t}} {dt}\:. \\ $$
Question Number 32718 Answers: 0 Comments: 0
$${find}\:\:\int_{\mathrm{0}} ^{\infty} \:{arctan}\left(\mathrm{2}{x}\right)\:\frac{{e}^{−{tx}} }{{x}}\:{dc}\:{with}\:{t}>\mathrm{0} \\ $$$$\left.\mathrm{2}\right)\:{calculate}\:\int_{\mathrm{0}} ^{\infty} \:\:\frac{{arctan}\left(\mathrm{2}{x}\right)}{{x}}\:{e}^{−{x}} \:{dx}. \\ $$
Question Number 32717 Answers: 0 Comments: 0
$${finf}\:\int_{\mathrm{0}} ^{+\infty} \:\:\:\frac{{dx}}{\mathrm{1}+{x}^{\mathrm{2}} \:+{x}^{\mathrm{4}} } \\ $$
Question Number 32716 Answers: 1 Comments: 0
$${find}\:\int_{\mathrm{0}} ^{\mathrm{2}\pi} \:\:\frac{{cos}^{\mathrm{2}} {x}}{\mathrm{1}+\mathrm{3}{sin}^{\mathrm{2}} {x}}{dx}\:. \\ $$
Question Number 32715 Answers: 0 Comments: 1
$${calculate}\:\int_{−\infty} ^{+\infty} \:\:\:\frac{{dt}}{\left(\mathrm{1}+{it}\right)\left(\mathrm{1}+{it}^{\mathrm{2}} \right)}\:\:. \\ $$
Question Number 32714 Answers: 0 Comments: 1
$${calculate}\:\:\int_{\mathrm{1}} ^{+\infty} \:\:\frac{{dt}}{{t}^{\mathrm{2}} \sqrt{\mathrm{1}+{t}^{\mathrm{2}} }}\:. \\ $$
Question Number 32712 Answers: 0 Comments: 1
$${calculate}\:\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \:\:\:\:\frac{{dt}}{\mathrm{1}+{a}\:{cos}^{\mathrm{2}} {t}}\:. \\ $$
Question Number 32722 Answers: 0 Comments: 0
$${find}\:\int_{\mathrm{0}} ^{\infty} \:\:\:\frac{{dx}}{\mathrm{1}+{x}^{\mathrm{3}} }\:. \\ $$
Question Number 32705 Answers: 0 Comments: 1
$${let}\:{give}\:\:{f}\left({x}\right)=\:\int_{\mathrm{0}} ^{\infty} \:\:{ln}\left(\mathrm{1}\:+\frac{{x}}{{t}^{\mathrm{2}} }\right){dt}\:{with}\:\mid{x}\mid<\mathrm{1} \\ $$$${find}\:{a}\:{simple}\:{form}\:{of}\:{f}\left({x}\right). \\ $$
Question Number 32704 Answers: 0 Comments: 0
$${find}\:\int_{\mathrm{0}} ^{\infty} \:\:\frac{\left({x}+\mathrm{1}\right)\sqrt{{x}}}{\mathrm{2}+{x}^{\mathrm{2}} }{dx}. \\ $$
Question Number 32675 Answers: 1 Comments: 1
Question Number 32708 Answers: 0 Comments: 1
$${let}\:{give}\:{f}\left({x}\right)=\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \:\frac{{ln}\left(\mathrm{1}+{xtant}\right)}{{tant}}{dt} \\ $$$${find}\:{a}\:{simple}\:{form}\:{of}\:{f}\left({x}\right) \\ $$$$\left.\mathrm{2}\right){calculate}\:\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \:\:\:\frac{{ln}\left(\mathrm{1}+\mathrm{2}{tant}\right)}{{tant}}{dt}\:. \\ $$
Question Number 32627 Answers: 0 Comments: 1
$${plzz}\:{help}\:{ne}\:{differentiate}\: \\ $$$${between} \\ $$$$\int{sin}\left(\mathrm{2}{x}\right)=\:−\frac{\mathrm{1}}{\mathrm{2}}{cox}\left(\mathrm{2}{x}\right)+{c}\: \\ $$$${is}\:{not}\:{change}\:{to}\:\int\mathrm{2}{sin}\left({x}\right){cos}\left({x}\right) \\ $$$${but}\:\underset{{b}} {\overset{{a}} {\int}}{sin}\left(\mathrm{2}{x}\right)=\:{is}\:{change}\:{to} \\ $$$$\underset{{b}} {\overset{{a}} {\int}}\mathrm{2}{sin}\left({x}\right){cos}\left({x}\right) \\ $$
Question Number 32484 Answers: 0 Comments: 2
$$ \\ $$$$\int_{\mathrm{1}} ^{\mathrm{2}} \int_{\mathrm{0}} ^{\mathrm{1}} \frac{{ln}\left({x}+{y}\right)}{\left({x}+{y}\right)}\:{dx}\:{dy} \\ $$
Question Number 32483 Answers: 0 Comments: 2
$${calculate}\:\int_{\mathrm{0}} ^{\mathrm{2}\pi} \:\:\:\:\frac{{dx}}{\mathrm{1}+\mathrm{2}{cosx}}\:. \\ $$
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