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Question Number 31088 Answers: 0 Comments: 0
$${find}\:\int_{\mathrm{0}} ^{\mathrm{1}} \:\:{dx}\:\int_{\mathrm{0}} ^{\mathrm{1}−{x}} \:\:{e}^{\frac{{y}−{x}}{{y}+{x}}} \:{dy}. \\ $$
Question Number 31087 Answers: 0 Comments: 0
$${find}\:\int\int\int_{{x}^{\mathrm{2}} \:+{y}^{\mathrm{2}} \:+{z}^{\mathrm{2}} \:<\mathrm{4}} \:\:\left({x}^{\mathrm{2}} \:+{y}^{\mathrm{2}} \:+{z}^{\mathrm{2}} \right){dxdydz}. \\ $$
Question Number 31086 Answers: 0 Comments: 0
$${find}\:\int\int_{{D}} \left({x}^{\mathrm{4}} \:−{y}^{\mathrm{4}} \right){dxdy}\:{with} \\ $$$${D}=\:\left\{\left({x},{y}\right)\in{R}^{\mathrm{2}} /\:\mathrm{1}<{x}^{\mathrm{2}} \:−{y}^{\mathrm{2}} <\mathrm{2}\:,\mathrm{1}<{xy}<\mathrm{2}\:,{x}>\mathrm{0},{y}>\mathrm{0}\right\} \\ $$
Question Number 31085 Answers: 0 Comments: 1
$${calculate}\:\int\int_{{x}^{\mathrm{2}} \:+{y}^{\mathrm{2}} \:−\mathrm{2}{x}\leqslant\mathrm{0}} {xdxdy}. \\ $$
Question Number 31084 Answers: 0 Comments: 1
$${find}\:\int\int_{{D}} \:\:\frac{{dxdy}}{\left({x}+{y}\right)^{\mathrm{4}} }\:\:{with}\:{D}=\left\{\left({x},{y}\right)\in{R}^{\mathrm{2}} /{x}\geqslant\mathrm{1},{y}\geqslant\mathrm{1},{x}+{y}\leqslant\mathrm{4}\right\} \\ $$
Question Number 31083 Answers: 0 Comments: 1
$${calculate}\:{by}\:{two}\:{methods}\:\int_{\mathrm{0}} ^{\mathrm{1}} \:\:\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \:\:\:\frac{{dx}\:{dt}}{\mathrm{1}+{x}^{\mathrm{2}} {tan}^{\mathrm{2}} {t}} \\ $$$${then}\:{find}\:{the}\:{value}\:{of}\:\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \:{t}\:{cotant}\:{dt}\:. \\ $$$$ \\ $$
Question Number 31082 Answers: 0 Comments: 0
$${calculate}\:{by}\:{two}\:{methods}\:\int_{\mathrm{0}} ^{\infty} \int_{\mathrm{0}} ^{\infty} \:\:\:\frac{{dxdy}}{\left(\mathrm{1}+{y}\right)\left(\mathrm{1}+{x}^{\mathrm{2}} {y}\right)} \\ $$$${then}\:{find}\:{the}\:{value}\:{of}\:\int_{\mathrm{0}} ^{\infty} \:\frac{{lnx}}{\mathrm{1}−{x}^{\mathrm{2}} }{dx}. \\ $$
Question Number 31081 Answers: 0 Comments: 0
$${find}\:\:\int_{\mathrm{0}} ^{\infty} {dx}\:\int_{{x}} ^{+\infty} \:{e}^{−{y}^{\mathrm{2}} {dy}} \:\:. \\ $$
Question Number 31080 Answers: 0 Comments: 0
$${find}\:\int_{\mathrm{0}} ^{\infty} \:{e}^{−{px}} {dx}\:\int_{\mathrm{0}} ^{{a}} \:\:\frac{{cos}\left({xt}\right)}{\sqrt{{a}^{\mathrm{2}} \:−{t}^{\mathrm{2}} }}{dt}\:{with}\:{a}>\mathrm{0}\:,{p}>\mathrm{0} \\ $$
Question Number 31079 Answers: 0 Comments: 0
$${calculate}\:\:\int\int_{\mathrm{0}\leqslant{x}\leqslant\mathrm{1}\:{and}\:\mathrm{0}\leqslant{y}\leqslant\mathrm{2}} \:\:\:{x}^{\mathrm{2}} {y}\:{e}^{{xy}} {dxdxy}. \\ $$
Question Number 31078 Answers: 0 Comments: 0
$${find}\:\int\int_{\mathrm{0}\leqslant{x}\leqslant\mathrm{3}\:{and}\:{x}\leqslant{y}\leqslant\mathrm{4}{x}−{x}^{\mathrm{2}} } \:\:\:\left({x}^{\mathrm{2}} \:+\mathrm{2}{y}\right){dxdy}. \\ $$
Question Number 31077 Answers: 0 Comments: 1
$${calculate}\:\int\int_{\mathrm{0}<{x}<\mathrm{1}{and}\:\mathrm{0}<{y}<{x}^{\mathrm{2}} } \:\frac{{y}}{\sqrt{{x}^{\mathrm{2}} \:+{y}^{\mathrm{2}} }}{dxdy}. \\ $$
Question Number 31076 Answers: 0 Comments: 1
$${find}\:\:\int_{\mathrm{0}} ^{\frac{\sqrt{\mathrm{2}}}{\mathrm{2}}} \:\:\:\:\:\:\:\frac{{dx}}{\left(\mathrm{2}{x}^{\mathrm{2}} \:+\mathrm{1}\right)\sqrt{\mathrm{1}+{x}^{\mathrm{2}} }}\:. \\ $$
Question Number 31075 Answers: 0 Comments: 0
$${find}\:\:\int_{\mathrm{0}} ^{\pi} \:\:\:\:\:\frac{{sin}\theta}{{cos}^{\mathrm{2}} \theta\:+\mathrm{2}\:{sin}^{\mathrm{2}} \theta}\:{d}\theta\:. \\ $$
Question Number 31074 Answers: 0 Comments: 1
$${find}\:\:\int_{{a}} ^{{b}} \:\sqrt{\left({b}−{x}\right)\left({x}−{a}\right)}\:{dx}\:{with}\:{a}<{b}\:.{then}\:{find}\: \\ $$$$\:\int_{\mathrm{1}} ^{\sqrt{\mathrm{2}}} \sqrt{\left(\sqrt{\mathrm{2}}\:−{x}\right)\left({x}−\mathrm{1}\right)}\:{dx}. \\ $$
Question Number 31073 Answers: 1 Comments: 1
$${find}\:{I}=\:\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \:\:\frac{\mathrm{1}−{sin}\theta}{{cos}\theta}{d}\theta\:. \\ $$
Question Number 31072 Answers: 0 Comments: 0
$${find}\:\:\int_{\mathrm{0}} ^{\infty} \:\:\:\:\frac{{dx}}{{e}^{{x}} \sqrt{{sh}\left(\mathrm{2}{x}\right)}}\:{dx}. \\ $$
Question Number 31071 Answers: 1 Comments: 3
$${find}\:\:\int_{\mathrm{0}} ^{\pi} \:\:\:\:\frac{{dx}}{\mathrm{1}+{sin}^{\mathrm{2}} {x}}\:. \\ $$
Question Number 31070 Answers: 0 Comments: 1
$${calculate}\:\int_{\mathrm{0}} ^{\pi} \:\:\:\:\:\frac{{dx}}{\mathrm{1}+\mathrm{2}{cosx}}\:. \\ $$
Question Number 31069 Answers: 1 Comments: 1
$${clculate}\:\:\int_{\mathrm{0}} ^{\mathrm{1}} \:{x}\sqrt{{x}^{\mathrm{2}} \:−\mathrm{2}{x}+\mathrm{2}}\:{dx} \\ $$
Question Number 31068 Answers: 0 Comments: 0
$${find}\:\:{I}_{{n}} =\int_{−\frac{\pi}{\mathrm{2}}} ^{\frac{\pi}{\mathrm{2}}} \:{e}^{−{ax}} \:{cos}^{\mathrm{2}{n}} {xdx}\:\:. \\ $$
Question Number 31067 Answers: 0 Comments: 0
$${find}\:{A}_{{n}} =\int_{\mathrm{0}} ^{\infty} \:{x}^{\mathrm{2}{n}} \:{e}^{−{ax}^{\mathrm{2}} } {dx}. \\ $$
Question Number 31066 Answers: 0 Comments: 0
$${find}\:\:{I}_{{n}} =\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \:{cos}^{\mathrm{2}{n}+\mathrm{1}} {xdx}. \\ $$
Question Number 31065 Answers: 0 Comments: 0
$${find}\:\:\int_{\mathrm{0}} ^{\pi} \:\:\:\frac{{xsinx}}{\left(\mathrm{1}−{acosx}\right)^{\mathrm{2}} }\:{dx}\:{with}\:\:\mid{a}\mid<\mathrm{1}. \\ $$
Question Number 31063 Answers: 0 Comments: 0
$${find}\:{f}\left({t}\right)=\:\int_{\mathrm{0}} ^{\mathrm{1}} \:{ln}\left(\mathrm{1}+{tx}^{\mathrm{2}} \right){dxfor}\:\:{t}>−\mathrm{1} \\ $$
Question Number 31062 Answers: 0 Comments: 0
$${find}\:\:\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \:{e}^{{x}} \:{sinx}\:{cos}^{\mathrm{2}} {xdx}. \\ $$
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