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IntegrationQuestion and Answers: Page 292
Question Number 36186 Answers: 0 Comments: 1
$${find}\:{nature}\:{of}\:\int_{\mathrm{1}} ^{+\infty} \sqrt{{t}}\:{sin}\left({t}^{\mathrm{2}} \right){dt}\:. \\ $$
Question Number 36185 Answers: 0 Comments: 2
$${study}\:{the}\:{vonvergence}\:{of}\: \\ $$$$\int_{\mathrm{1}} ^{+\infty} \:\:\frac{{e}^{−\frac{\mathrm{1}}{{t}}} \:−{cos}\left(\frac{\mathrm{1}}{{t}}\right)}{{t}}{dt} \\ $$
Question Number 36184 Answers: 0 Comments: 1
$${study}\:{the}\:{convergence}\:{of}\:\:\int_{\mathrm{1}} ^{+\infty} \:\frac{{cos}\left({t}\right)}{\sqrt{{t}}}{dt} \\ $$
Question Number 36183 Answers: 0 Comments: 0
$${calculate}\:\:\int_{\mathrm{1}} ^{+\infty} \:\:{arctan}\left(\frac{\mathrm{1}}{{t}}\right){dt} \\ $$
Question Number 36182 Answers: 2 Comments: 1
$${calculate}\:\:\int_{\mathrm{1}} ^{+\infty} \:\:\:\frac{{dt}}{{t}\sqrt{\mathrm{1}+{t}^{\mathrm{2}} }} \\ $$
Question Number 36181 Answers: 0 Comments: 1
$${let}\:{I}\left(\xi\right)\:\:=\:\int_{\xi} ^{\mathrm{1}−\xi} \:\:\:\frac{{dt}}{\mathrm{1}−\left({t}−\xi\right)^{\mathrm{2}} } \\ $$$${find}\:{lim}_{\xi\rightarrow\mathrm{0}^{+} } \:\:\:{I}\left(\xi\right) \\ $$
Question Number 36180 Answers: 1 Comments: 1
$${calculate}\:\:\int_{\mathrm{0}} ^{\mathrm{1}} \:\:\frac{{ln}\left({t}\right)}{\left(\mathrm{1}+{t}\right)^{\mathrm{2}} }{dt} \\ $$
Question Number 36167 Answers: 0 Comments: 2
$${let}\:{give}\:\:{I}\:\:=\:\int_{\mathrm{0}} ^{\infty} \:\:\:\frac{{dx}}{\left({x}^{\mathrm{2}} \:+{i}\right)^{\mathrm{2}} } \\ $$$$\left.\mathrm{1}\right)\:{extract}\:{Re}\left({I}\right)\:{and}\:{Im}\left({I}\right) \\ $$$$\left.\mathrm{2}\right)\:{find}\:{the}\:{value}\:{of}\:{I} \\ $$$$\left.\mathrm{3}\right)\:{calculate}\:{Re}\left({I}\right)\:{and}\:{Im}\left({I}\right)\:. \\ $$
Question Number 36057 Answers: 2 Comments: 1
$${find}\:{the}\:{value}\:{of}\:\:\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{4}}} \:\:\:\:\frac{{cosx}}{{sinx}\:+{tanx}}{dx}\: \\ $$
Question Number 36056 Answers: 1 Comments: 2
$${calculate}\:\:\int_{−\infty} ^{+\infty} \:\:\:\frac{\mathrm{2}{x}}{\left({x}^{\mathrm{2}} \:+{mx}\:+\mathrm{1}\right)^{\mathrm{2}} }{dx}\:{with}\:\mid{m}\mid<\mathrm{2} \\ $$
Question Number 36031 Answers: 0 Comments: 0
$$\mathrm{Q}.\:\mathrm{Evaluate}:\:\:\:\int_{\int\mathrm{xyzdxdydz}} ^{\int\mathrm{zyxdzdydx}} \int_{\frac{\mathrm{d}}{\mathrm{dx}}\left(\mathrm{x}^{\mathrm{sin}\:\mathrm{x}} \right)} ^{\frac{\mathrm{d}}{\mathrm{dx}}\left(\mathrm{x}^{\mathrm{cos}\:\mathrm{x}} \right)} \int_{\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\frac{−\mathrm{x}^{\mathrm{2}} +\mathrm{2}}{\mathrm{x}}} ^{\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\frac{\mathrm{x}^{\mathrm{2}} −\mathrm{2}}{\mathrm{x}}} \int_{\mathrm{0}} ^{\infty} \mathrm{w}^{\mathrm{1}−\mathrm{x}} \mathrm{x}^{\mathrm{1}−\mathrm{y}} \mathrm{y}^{\mathrm{1}−\mathrm{z}} \mathrm{z}^{\mathrm{1}−\mathrm{w}} \mathrm{dwdxdydz} \\ $$
Question Number 36009 Answers: 0 Comments: 1
$${calculate}\:\int_{−\infty} ^{+\infty} \:\:\:\:\:\frac{{xdx}}{\left(\mathrm{2}{x}+\mathrm{1}+{i}\right)^{\mathrm{3}} }\:\:{with}\:{i}^{\mathrm{2}} \:=−\mathrm{1}\:. \\ $$
Question Number 35990 Answers: 0 Comments: 2
$${calculate}\:\int_{\mathrm{2}} ^{\mathrm{5}} \:\:\:\:\frac{{xdx}}{\mathrm{2}{x}+\mathrm{1}\:+\sqrt{{x}−\mathrm{1}}} \\ $$
Question Number 35983 Answers: 0 Comments: 1
$${calculate}\:\:\int_{\mathrm{0}} ^{\infty} \:\:\:\:\:\frac{\mathrm{2}\sqrt{{t}}\:+\mathrm{1}}{{t}^{\mathrm{5}} \:\:\:+\mathrm{3}}{dt}\:\:. \\ $$
Question Number 35982 Answers: 0 Comments: 0
$${let}\:{f}\left({t}\right)\:=\int_{\mathrm{0}} ^{\infty} \:\:\:{e}^{−{arctsn}\left(\:\mathrm{1}+{tx}^{\mathrm{2}} \right)} {dx}\:\:{with}\:{t}\:{from}\:{R} \\ $$$$\left.\mathrm{1}\right)\:{calculate}\:{f}^{'} \left({t}\right) \\ $$$$\left.\mathrm{2}\right)\:{find}\:{a}\:{simple}\:{form}\:{of}\:{f}\left({t}\right)\:. \\ $$
Question Number 35949 Answers: 1 Comments: 1
$$\int_{\mathrm{0}} ^{\:\:\alpha} \frac{\mathrm{tan}\:\theta}{\sqrt{{a}^{\mathrm{2}} \mathrm{cos}\:^{\mathrm{2}} \theta−{b}^{\mathrm{2}} \mathrm{sin}\:^{\mathrm{2}} \theta}}\:{d}\theta\:=\:? \\ $$
Question Number 35920 Answers: 1 Comments: 1
$$\int\:\frac{{e}^{\mathrm{2}{x}} +\mathrm{1}}{\mathrm{2}{e}^{{x}} −\mathrm{1}}\:{dx}\:=\:? \\ $$
Question Number 35909 Answers: 1 Comments: 4
$$\int\frac{\mathrm{7}{x}−\mathrm{6}}{\left({x}^{\mathrm{2}} +\mathrm{25}\right)\sqrt{\left({x}−\mathrm{3}\right)^{\mathrm{2}} +\mathrm{4}}}\:{dx}\:=\:? \\ $$
Question Number 35832 Answers: 1 Comments: 3
$${find}\:{the}\:{value}\:{of}\:\:{f}\left(\lambda\right)\:=\:\int_{−{a}} ^{{a}} \:\:\:\frac{{dx}}{\left(\lambda\:+_{} {x}^{\mathrm{2}} \right)^{\frac{\mathrm{3}}{\mathrm{2}}} } \\ $$$$\lambda\in{R}\:. \\ $$
Question Number 35821 Answers: 0 Comments: 4
$${let}\:{f}\left({t}\right)\:=\:\int_{\mathrm{0}} ^{\infty} \:\:\frac{{e}^{−{ax}} \:−{e}^{−{bx}} }{{x}^{\mathrm{2}} }\:{e}^{−{tx}^{\mathrm{2}} } {dx}\:\:\:{with}\:{t}>\mathrm{0} \\ $$$$\left.\mathrm{1}\right)\:{calculate}\:{f}^{'} \left({t}\right) \\ $$$$\left.\mathrm{2}\right){find}\:{a}\:{simple}\:{form}\:{of}\:{f}\left({t}\right) \\ $$$$\left.\mathrm{3}\right)\:{find}\:{the}\:{value}\:{of}\:\int_{\mathrm{0}} ^{\infty} \:\:\frac{{e}^{−\mathrm{2}{x}} \:\:−{e}^{−{x}} }{{x}^{\mathrm{2}} }\:{e}^{−\mathrm{3}{x}^{\mathrm{2}} } {dx} \\ $$
Question Number 35732 Answers: 1 Comments: 1
Question Number 35729 Answers: 1 Comments: 1
$${find}\:{the}\:{value}\:{of}\:\:\:{I}\:=\int_{\mathrm{0}} ^{\pi} \:\:\:\:\frac{{dx}}{\mathrm{2}−{cosx}} \\ $$
Question Number 35691 Answers: 0 Comments: 0
$${calculate}\:{lim}_{{a}\rightarrow\mathrm{0}^{+} \:\:\:\:} \:\:\:\int_{−{a}} ^{{a}} \:\:\sqrt{\frac{\mathrm{1}+{x}^{\mathrm{2}} }{{a}^{\mathrm{2}} \:−{x}^{\mathrm{2}} }}\:\:{dx}\:. \\ $$
Question Number 35687 Answers: 1 Comments: 2
$${calculate}\:{f}\left({a}\right)=\int_{\mathrm{0}} ^{\pi} \:\:\:\:\:\frac{{dx}}{\mathrm{1}−{a}\:{cosx}}\:\:{a}\:{from}\:{R}\:. \\ $$$$\left.\mathrm{2}\right)\:{application}\:\:{calculate}\:\:\int_{\mathrm{0}} ^{\pi} \:\:\:\:\frac{{dx}}{\mathrm{1}−\mathrm{2}{cosx}} \\ $$
Question Number 35686 Answers: 1 Comments: 1
$${calculate}\:\:\int_{\sqrt{\mathrm{3}}} ^{+\infty} \:\:\:\:\:\frac{{dx}}{{x}\sqrt{\:\mathrm{2}+{x}^{\mathrm{2}} }}\:. \\ $$
Question Number 35685 Answers: 1 Comments: 1
$${calculate}\:\:\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{4}}} \:\:\:{x}\:{artan}\left(\mathrm{2}{x}+\mathrm{1}\right){dx} \\ $$
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