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IntegrationQuestion and Answers: Page 292
Question Number 33338 Answers: 0 Comments: 1
$${find}\:{lim}_{{n}\rightarrow\infty} \:\int_{\mathrm{0}} ^{+\infty} \:{tan}^{{n}} {xdx}\:.\: \\ $$
Question Number 33333 Answers: 0 Comments: 0
$${let}\:{hive}\:\:{I}_{{n}} =\:\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \:\left({sinx}\right)^{{n}} \:{dx} \\ $$$${prove}\:{that}\:\:{I}_{{n}} \:\sim\:\:\sqrt{\frac{\pi}{\mathrm{2}{n}}}\:\left({n}\rightarrow\infty\right) \\ $$$$ \\ $$
Question Number 33334 Answers: 0 Comments: 3
$${decompose}\:{F}\left({x}\right)=\:\frac{{x}^{\mathrm{2}} }{{x}^{\mathrm{4}} \:−\mathrm{1}}\:{imside}\:{R}\left({x}\right)\: \\ $$$$\left.\mathrm{2}\right)\:{find}\:{the}\:{value}\:{of}\:\:\:\:\int_{\mathrm{2}} ^{+\infty} \:\:\frac{{x}^{\mathrm{2}} }{{x}^{\mathrm{4}} \:−\mathrm{1}}\:\:. \\ $$
Question Number 33331 Answers: 0 Comments: 1
$${calcilate}\:\:\int_{\mathrm{0}} ^{\mathrm{1}} \:\:\:\:\frac{{dx}}{\left(\mathrm{1}+{x}^{\mathrm{2}} \right)^{\mathrm{3}} } \\ $$
Question Number 33329 Answers: 0 Comments: 1
$${find}\:\:\int\:\:\:\frac{{dx}}{\sqrt{\mathrm{4}{x}−{x}^{\mathrm{2}} }}\:. \\ $$
Question Number 33328 Answers: 0 Comments: 1
$${find}\:\:\:\int_{\frac{\pi}{\mathrm{4}}} ^{\frac{\mathrm{4}}{\pi}} \:\:\left(\mathrm{1}+\frac{\mathrm{1}}{{x}^{\mathrm{2}} }\right){arctanx}\:{dx} \\ $$
Question Number 33327 Answers: 0 Comments: 1
$${find}\:\:\int_{\mathrm{0}} ^{\mathrm{1}} \:\:\:\frac{{dx}}{\mathrm{3}\:+{e}^{−{x}} } \\ $$
Question Number 33311 Answers: 0 Comments: 0
$${let}\:\:{f}\left({x}\right)\:=\mid{sinx}\mid\:\:\left(\mathrm{2}\pi\:{periodic}\:{even}\right) \\ $$$${developp}\:{f}\:{at}\:{fourier}\:{serie} \\ $$
Question Number 33310 Answers: 0 Comments: 0
$${let}\:{consider}\:{the}\:\mathrm{2}\pi\:{periodic}?{function}\:\:{f}\left({x}\right)\:={e}^{{x}} \\ $$$$\left.\mathrm{1}\right)\:{developp}\:{f}\:{at}\:{fourier}\:{serie} \\ $$$$\left.\mathrm{2}\right)\:{find}\:{the}\:{value}\:{of}\:\:\sum_{{n}=\mathrm{0}} ^{\infty} \:\:\frac{\left(−\mathrm{1}\right)^{{n}} }{{n}^{\mathrm{2}} \:+\mathrm{1}} \\ $$
Question Number 33297 Answers: 0 Comments: 0
$${find}\:\:\:\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \:{ln}\left(\mathrm{1}+{x}\:{sin}\theta\right){d}\theta\:\:\:{with}\:\:\mathrm{0}<{x}<\mathrm{1} \\ $$$$\left.\mathrm{2}\right)\:{calculate}\:\:\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \:\:{ln}\left(\mathrm{1}+\frac{\mathrm{1}}{\mathrm{2}}{sin}\theta\right){d}\theta \\ $$
Question Number 33259 Answers: 0 Comments: 1
$${find}\:{the}\:{value}\:{of}\:\:\int_{\mathrm{0}} ^{\infty} \:\:\:\frac{{arctan}\left(\mathrm{2}{x}\right)}{{a}^{\mathrm{2}} \:+{x}^{\mathrm{2}} }\:{dx}\:{with}\:{a}\neq\mathrm{0} \\ $$
Question Number 33258 Answers: 0 Comments: 0
$${if}\:\:\:\frac{\mathrm{1}}{\mathrm{1}+{cosx}}\:=\:\frac{{a}_{\mathrm{0}} }{\mathrm{2}}\:+\sum_{{n}\geqslant\mathrm{1}} {a}_{{n}} {cos}\left({nx}\right)\:{calculate}\:{a}_{\mathrm{0}} \\ $$$${and}\:{a}_{{n}} \\ $$
Question Number 33257 Answers: 0 Comments: 1
$${let}\:{g}\left({x}\right)=\:\frac{\mathrm{1}}{\mathrm{1}+{x}^{\mathrm{4}} } \\ $$$$\left.\mathrm{1}\right)\:{find}\:{g}^{\left({n}\right)} \left({x}\right) \\ $$$$\left.\mathrm{2}\right)\:{calculate}\:{g}^{\left({n}\right)} \left(\mathrm{0}\right) \\ $$$$\left.\mathrm{3}\right)\:{if}\:{g}\left({x}\right)=\Sigma\:{u}_{{n}} \:{x}^{{n}} \:\:\:{find}\:{the}\:{sequence}\:{u}_{{n}} \\ $$
Question Number 33222 Answers: 0 Comments: 0
$${let}\:{give}\:{n}\:\geqslant\mathrm{3}\:{integr}\:\:{calculate} \\ $$$${I}_{{n}} =\:\int_{−\infty} ^{+\infty} \:\:\:\:\:\frac{{dx}}{\mathrm{1}+{x}\:+{x}^{\mathrm{2}} \:+....+{x}^{{n}−\mathrm{1}} } \\ $$
Question Number 33223 Answers: 0 Comments: 0
$${let}\:\:{A}_{{n}} \:=\int_{−\infty} ^{+\infty} \:\:\:\frac{{e}^{{i}\pi{x}} }{\mathrm{1}+{x}+{x}^{\mathrm{2}} \:+...{x}^{{n}−\mathrm{1}} }\:\:{with}\:{n}\geqslant\mathrm{3}\:{integr} \\ $$$${find}\:{the}\:{value}\:{of}\:{A}_{{n}} \:. \\ $$
Question Number 33210 Answers: 0 Comments: 0
$${find}\:\:{lim}_{{x}\rightarrow+\infty} \:\:{x}\:{e}^{−{x}^{\mathrm{2}} } \:\:\:\underset{\mathrm{0}} {\int}^{{x}−\mathrm{1}} \:\:{e}^{{t}^{\mathrm{2}} } \:{dt} \\ $$
Question Number 33204 Answers: 0 Comments: 1
$${find}\:{the}\:{value}\:{of}\:\int_{−\infty} ^{+\infty} \:\:\:\:\frac{{cos}\left({ax}\right)}{\mathrm{1}+{x}+{x}^{\mathrm{2}} }\:{dx}. \\ $$
Question Number 33232 Answers: 0 Comments: 1
$${find}\:{the}\:{value}\:{of}\:\:\:\int_{−\infty} ^{+\infty} \:\:\:\frac{{x}\:{sin}\left(\mathrm{2}{x}\right)}{\left(\mathrm{1}+\mathrm{4}{x}^{\mathrm{2}} \right)^{\mathrm{2}} }\:{dx}\:. \\ $$
Question Number 33202 Answers: 0 Comments: 1
$${find}\:{the}\:{value}\:{of}\:\:\int_{−\infty} ^{+\infty} \:\:\:\:\:\frac{{dt}}{\left(\mathrm{1}+{t}\:+{t}^{\mathrm{2}} \right)^{\mathrm{2}} }\:. \\ $$
Question Number 33175 Answers: 0 Comments: 1
$${find}\:\int_{\mathrm{0}} ^{\mathrm{1}} \:\:\frac{{dt}}{\left(\mathrm{1}+{t}^{\mathrm{2}} \right)^{\mathrm{2}} } \\ $$
Question Number 33172 Answers: 0 Comments: 0
$${find}\:\:\int\:\:\:\:\frac{{dx}}{{x}\:+\sqrt{{x}^{\mathrm{2}} \:−\mathrm{3}{x}+\mathrm{2}}}\:. \\ $$
Question Number 33170 Answers: 0 Comments: 1
$${prove}\:{that}\:\:\int_{\mathrm{0}} ^{\infty} \:\frac{\mid{sinx}\mid}{{x}}\:{dx}\:{is}\:{divergent}. \\ $$
Question Number 33169 Answers: 1 Comments: 1
$${find}\:{the}\:{value}\:{of}\:\:\int_{\mathrm{0}} ^{\pi} \:\:\:\:\frac{{dx}}{\mathrm{1}+\mathrm{2}\:{sin}^{\mathrm{2}} {x}}\:\:. \\ $$
Question Number 33166 Answers: 0 Comments: 0
$${find}\:{the}\:{value}\:{of}\:\:\int_{\mathrm{0}} ^{\mathrm{1}} \:\:\:\frac{{dx}}{\mathrm{1}+{x}^{\mathrm{4}} }\:\:. \\ $$
Question Number 33155 Answers: 0 Comments: 4
$$\mathrm{Evaluate} \\ $$$$\int_{−\infty} ^{\infty} \:\mathrm{3}{x}^{\mathrm{2}} \left({x}^{\mathrm{3}} \:+\:\mathrm{1}\right)^{\mathrm{2}} \:{e}^{−{x}^{\mathrm{6}} \:−\:\mathrm{2}{x}^{\mathrm{3}} } \:{dx} \\ $$
Question Number 33130 Answers: 0 Comments: 0
$${find}\:\:{the}\:{value}\:{of}\:\:\int_{\mathrm{0}} ^{\infty} \:\:\frac{\mathrm{1}+{x}\:{cos}\theta}{{x}^{\mathrm{2}} \:+\mathrm{2}{x}\:{cos}\theta\:+\mathrm{1}}\:{dx}\:. \\ $$
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