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IntegrationQuestion and Answers: Page 290
Question Number 35062 Answers: 0 Comments: 0
$${calculate}\:{A}_{{n}} \:=\:\int_{\mathrm{0}} ^{\infty} \:\:\:\:\:\frac{{dt}}{\left({t}^{\mathrm{4}} \:+\mathrm{1}\right)^{{n}} } \\ $$$${with}\:{n}\:{integr}\:{natural}\:. \\ $$
Question Number 35061 Answers: 2 Comments: 1
$${find}\:\:\:\int_{\mathrm{0}} ^{\infty} \:\:\:\:\:\:\:\frac{{x}^{\mathrm{2}} \:+\mathrm{3}}{\left({x}^{\mathrm{4}} \:+\mathrm{1}\right)^{\mathrm{2}} }{dx} \\ $$
Question Number 35060 Answers: 1 Comments: 1
$${calculate}\:\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{4}}} \:{sinx}\:{ln}\left({cosx}\right){dx} \\ $$
Question Number 35059 Answers: 2 Comments: 2
$${find}\:\:\:\int_{\mathrm{0}} ^{\pi} \:\:\:\:\:\:\frac{{dx}}{{cosx}\:+{sinx}} \\ $$
Question Number 35058 Answers: 1 Comments: 1
$${find}\:\:\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{4}}} \:\:\:\:\frac{{dt}}{\left(\mathrm{1}+{cos}^{\mathrm{2}} {t}\right)^{\mathrm{3}} } \\ $$
Question Number 35055 Answers: 1 Comments: 2
$${calculate}\:\int_{−\infty} ^{+\infty} \:\:\:\:\:\frac{{dx}}{\left(\mathrm{1}+{x}+{x}^{\mathrm{2}} \right)^{\mathrm{3}} } \\ $$
Question Number 35054 Answers: 0 Comments: 1
$${find}\:\:\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{4}}} \:\:\:\:\frac{{xdx}}{\mathrm{2}\:+{cosx}} \\ $$
Question Number 35053 Answers: 0 Comments: 0
$${let}\:{v}\left({x}\right)={ln}\left(\mathrm{1}+{x}+{x}^{\mathrm{2}} \right) \\ $$$${developp}\:{f}\:{at}\:{integr}\:{serie}. \\ $$
Question Number 35049 Answers: 1 Comments: 1
$${let}\:{A}_{{n}} \:=\:\int_{\frac{\mathrm{1}}{{n}}} ^{{n}} \:\left(\mathrm{1}+\frac{\mathrm{1}}{{x}^{\mathrm{2}} }\right){arctanx}\:{dx} \\ $$$$\left.\mathrm{1}\right)\:{calculate}\:{A}_{{n}} \\ $$$$\left.\mathrm{2}\right)\:{find}\:{lim}_{{n}\rightarrow+\infty} \:{A}_{{n}} \\ $$
Question Number 35048 Answers: 0 Comments: 0
$${find}\:\int\:\:\:\:\:\frac{{dx}}{{cos}\left({sinx}\right)} \\ $$
Question Number 35046 Answers: 0 Comments: 0
$${find}\:{F}\left({x}\right)=\:\int_{\mathrm{0}} ^{\pi} \:{ln}\left(\:\mathrm{1}+{x}\:{sin}^{\mathrm{2}} {t}\right){dt}\:{with}\:\mid{x}\mid<\mathrm{1} \\ $$$$\left.\mathrm{2}\right)\:{calculate}\:\int_{\mathrm{0}} ^{\pi} \:\:{ln}\left(\mathrm{1}+\frac{\mathrm{1}}{\mathrm{2}}{sin}^{\mathrm{2}} {t}\right){dt} \\ $$
Question Number 35045 Answers: 0 Comments: 0
$${find}\:{f}\left({x}\right)=\int_{\mathrm{0}} ^{\infty} \:\:\:\frac{{arctan}\left({xt}\right)}{\mathrm{1}+{t}^{\mathrm{2}} }{dt}\:\:. \\ $$
Question Number 35044 Answers: 1 Comments: 1
$$\left.\mathrm{1}\right){find}\:\int\:\sqrt{\mathrm{1}+{t}^{\mathrm{2}} }\:{dt} \\ $$$$\left.\mathrm{2}\right)\:{calculate}\:\:\int_{\mathrm{1}} ^{\sqrt{\mathrm{3}}} \:\sqrt{\mathrm{1}+{t}^{\mathrm{2}} }\:\:{dt} \\ $$
Question Number 35043 Answers: 1 Comments: 0
$${let}\:{t}>\mathrm{0}\:{and}\:{F}\left({t}\right)\:=\int_{\mathrm{0}} ^{\infty} \:\:\:\frac{{sin}\left({x}^{\mathrm{2}} \right)\:{e}^{−{tx}^{\mathrm{2}} } }{{x}^{\mathrm{2}} }{dx} \\ $$$${calculate}\:\frac{{dF}}{{dt}}\left({t}\right). \\ $$
Question Number 35018 Answers: 1 Comments: 0
$$\int\int\int\frac{{dxdydz}}{\left({x}+{y}+{z}+\mathrm{1}\right)^{\mathrm{3}} }\:\:\:{bounded}\:{by}\:{the} \\ $$$${coordinate}\:{planes}\:{and}\:{the}\:{plane} \\ $$$${x}+{y}+{z}=\mathrm{1}\:. \\ $$
Question Number 35015 Answers: 2 Comments: 0
$$\int\frac{{x}^{\mathrm{2}} }{\left(\mathrm{1}+{x}^{\mathrm{3}} \right)^{\mathrm{2}} }{dx} \\ $$
Question Number 34992 Answers: 1 Comments: 1
$$\:\underset{\mathrm{0}} {\overset{\pi} {\int}}\frac{\mathrm{cos}\left({x}\right)}{\mathrm{1}+\mathrm{2sin}\left(\mathrm{2}{x}\right)}{dx} \\ $$
Question Number 34956 Answers: 3 Comments: 2
Question Number 34911 Answers: 1 Comments: 1
$${find}\:\:\:\int_{\mathrm{2}} ^{\mathrm{3}} \:\:\:\:\frac{\mathrm{2}{x}^{\mathrm{2}} \:+\mathrm{3}}{\left({x}−\mathrm{1}\right)^{\mathrm{2}} \left({x}^{\mathrm{2}} +\mathrm{1}\right)}\:{dx} \\ $$
Question Number 34910 Answers: 0 Comments: 1
$${find}\:{J}_{{n},{p}} \:=\int_{\mathrm{0}} ^{\infty} \:\:{x}^{{n}} \:\:{e}^{−\frac{{x}^{\mathrm{2}} }{{p}}} \:\:{dx}\:\:{with}\:{p}>\mathrm{0}\:{and}\:{n}\:{integr} \\ $$
Question Number 34901 Answers: 0 Comments: 3
$$\int_{−\pi/\mathrm{2}} ^{+\pi/\mathrm{2}} \sqrt{\mathrm{cos}^{\mathrm{2}{n}−\mathrm{1}} {x}−\mathrm{cos}^{\mathrm{2}{n}+\mathrm{1}} {x}}{dx} \\ $$$$=\left[−\frac{\mathrm{2cos}^{\frac{\mathrm{2}{n}+\mathrm{1}}{\mathrm{2}}} {x}}{\mathrm{2}{n}+\mathrm{1}}\right]_{−\pi/\mathrm{2}} ^{+\pi/\mathrm{2}} =\mathrm{0}? \\ $$$$\mathrm{What}\:\mathrm{is}\:\mathrm{the}\:\mathrm{mistake}\:\mathrm{in}\:\mathrm{above}? \\ $$$$\int_{−\pi/\mathrm{2}} ^{+\pi/\mathrm{2}} \sqrt{\mathrm{cos}^{\mathrm{2}{n}−\mathrm{1}} {x}−\mathrm{cos}^{\mathrm{2}{n}+\mathrm{1}} {x}}{dx} \\ $$$$=\mathrm{2}\int_{\mathrm{0}} ^{\pi/\mathrm{2}} \sqrt{\mathrm{cos}^{\mathrm{2}{n}−\mathrm{1}} {x}−\mathrm{cos}^{\mathrm{2}{n}+\mathrm{1}} {x}}{dx} \\ $$$$=\frac{\mathrm{4}}{\mathrm{2}{n}+\mathrm{1}}\:\left(\mathrm{this}\:\mathrm{is}\:\mathrm{correct}\:\mathrm{answer}\right) \\ $$
Question Number 34866 Answers: 0 Comments: 0
$${find}\:{f}\left({x}\right)=\int_{\mathrm{0}} ^{\infty} \:\frac{{arctan}\left({x}\left({t}\:+\frac{\mathrm{1}}{{t}}\right)\right)}{\mathrm{1}+{t}^{\mathrm{2}} }{dt}\: \\ $$
Question Number 34862 Answers: 2 Comments: 8
$${find}\:{the}\:{value}\:{of} \\ $$$${f}\left({x}\right)\:=\:\int_{\mathrm{0}} ^{\pi} \:\:\:\frac{{cosx}}{\mathrm{1}+\mathrm{2}{sin}\left(\mathrm{2}{x}\right)}{dx} \\ $$
Question Number 34827 Answers: 1 Comments: 5
$$\boldsymbol{{Find}}\:\int\:\boldsymbol{{Sin}}^{\mathrm{6}} \boldsymbol{{x}}\:\boldsymbol{{dx}} \\ $$$$ \\ $$
Question Number 34771 Answers: 0 Comments: 1
$${let}\:{A}\left({x}\right)=\:\int_{\mathrm{0}} ^{\mathrm{1}} \:{ln}\left(\mathrm{1}+{ix}^{\mathrm{2}} \right){dx} \\ $$$${find}\:{a}\:{simple}\:{form}\:{of}\:{f}\left({x}\right)\:\:\:\:\left({x}\in{R}\right) \\ $$
Question Number 34720 Answers: 0 Comments: 0
$${let}\:{B}\left({p},{q}\right)\:=\:\int_{\mathrm{0}} ^{\mathrm{1}} \:{x}^{{p}−\mathrm{1}} \left(\mathrm{1}−{x}\right)^{{q}−\mathrm{1}} {dx} \\ $$$${calculate}\:{B}\left(\frac{\mathrm{1}}{\mathrm{3}},\:\frac{\mathrm{1}}{\mathrm{3}}\right) \\ $$$$\left.\mathrm{2}\right)\:{calculate}\:{B}\left(\frac{\mathrm{1}}{\mathrm{2}}\:,\frac{\mathrm{2}}{\mathrm{3}}\right)\:. \\ $$
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