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IntegrationQuestion and Answers: Page 263
Question Number 43027 Answers: 2 Comments: 1
Question Number 43008 Answers: 1 Comments: 0
$$\left({y}'\right)^{\mathrm{2}} =−\mathrm{1}+\mathrm{sin}\:{x} \\ $$$${y}=? \\ $$
Question Number 42994 Answers: 0 Comments: 5
$$\int\sqrt{\mathrm{1}+\frac{\mathrm{cos}\:{x}}{\mathrm{4tan}\:{x}}}{dx}=? \\ $$
Question Number 42945 Answers: 2 Comments: 12
$$\int_{\mathrm{0}} ^{\:\:\pi/\mathrm{2}} \frac{{dx}}{\sqrt{\mathrm{sin}\:{x}}}\:=\:? \\ $$
Question Number 42870 Answers: 0 Comments: 0
$${let}\:\mathrm{0}<{x}<\mathrm{1}\:\:{and}\:\Gamma\left({x}\right)\:=\int_{\mathrm{0}} ^{\infty} \:{t}^{{x}−\mathrm{1}} \:{e}^{−{t}} \:{dt}\: \\ $$$$\left.\mathrm{1}\right)\:{prove}\:{that}\:\Gamma\left({x}\right).\Gamma\left(\mathrm{1}−{x}\right)\:=\frac{\pi}{{sin}\left(\pi{x}\right)}\:\:\:\left({compliments}\:{formulae}\right) \\ $$$$\left.\mathrm{2}\right)\:{calculate}\:\Gamma\left({n}\right)\:{and}\:\Gamma\left({n}+\frac{\mathrm{1}}{\mathrm{2}}\right)\:{with}\:{n}\:{from}\:{N}. \\ $$
Question Number 42823 Answers: 1 Comments: 1
$$\mathrm{Evaluate}\:: \\ $$$$\int_{−\mathrm{5}} ^{\:\mathrm{5}} \:{x}^{\mathrm{2}} \left[{x}+\frac{\mathrm{1}}{\mathrm{2}}\right]{dx}\:=\:\:? \\ $$$${where}\:\left[.\right]=\:{greatest}\:{integer}\:{function} \\ $$
Question Number 42812 Answers: 0 Comments: 1
$${study}\:{the}\:{convervence}\:{of}\:\:\int_{\mathrm{1}} ^{+\infty} \:\:\frac{{arctan}\left({x}−\mathrm{1}\right)}{\left({x}^{\mathrm{2}} −\mathrm{1}\right)^{\frac{\mathrm{4}}{\mathrm{3}}} }\:{dx} \\ $$
Question Number 42810 Answers: 1 Comments: 1
$${find}\:\:\int_{\mathrm{0}} ^{\infty} \:\:\:\:\frac{{x}^{\mathrm{5}} }{\mathrm{1}+{x}^{\mathrm{7}} }{dx}\:\:. \\ $$
Question Number 42809 Answers: 1 Comments: 1
$${calculate}\:\int_{\mathrm{0}} ^{\infty} \:\:\:\frac{{tdt}}{\left(\mathrm{1}+{t}^{\mathrm{4}} \right)^{\mathrm{2}} } \\ $$
Question Number 42806 Answers: 0 Comments: 0
$${let}\:\:{u}_{{n}} =\:\int_{{n}} ^{{n}+\mathrm{2}} \:\:\:\frac{\left({t}+{n}\right)^{\frac{\mathrm{1}}{\mathrm{4}}} }{{t}^{\frac{\mathrm{1}}{\mathrm{3}}} }{dt} \\ $$$${find}\:{lim}_{{n}\rightarrow+\infty} \:{u}_{{n}} \\ $$
Question Number 42804 Answers: 1 Comments: 1
$${calculate}\:\:\int_{\frac{\mathrm{1}}{\mathrm{2}}} ^{\mathrm{1}} \:\:\:\:\:\frac{{dx}}{\sqrt{\mathrm{4}{x}^{\mathrm{2}} \:−\mathrm{1}}\:+\sqrt{\mathrm{4}{x}^{\mathrm{2}} \:+\mathrm{1}}} \\ $$
Question Number 42803 Answers: 1 Comments: 0
$${find}\:\:\int_{\mathrm{0}} ^{\mathrm{1}} \:\left({x}^{\mathrm{2}} \:+\mathrm{1}\right)\sqrt{\frac{\mathrm{1}−{x}}{\mathrm{1}+{x}}}{dx}\: \\ $$
Question Number 42802 Answers: 0 Comments: 1
$${calculate}\:\:\int_{\frac{\mathrm{1}}{\mathrm{2}}} ^{\frac{\mathrm{5}}{\mathrm{4}}} \:\:\:\frac{{x}^{\mathrm{3}} }{\sqrt{\mathrm{2}+{x}−{x}^{\mathrm{2}} }}{dx} \\ $$
Question Number 42801 Answers: 1 Comments: 1
$${find}\:{f}\left({x}\right)\:=\:\int_{\frac{\pi}{\mathrm{4}}} ^{\frac{\pi}{\mathrm{3}}} \:\:\:\:\frac{{cosxdx}}{\mathrm{2}{cos}^{\mathrm{2}} {x}\:+{sin}^{\mathrm{2}} {x}\:+\mathrm{1}} \\ $$
Question Number 42800 Answers: 1 Comments: 0
$${find}\:\int\:\:\:\:\:\frac{{sinx}}{\mathrm{1}+\mathrm{2}\:{cosx}}{dx} \\ $$
Question Number 42799 Answers: 0 Comments: 2
$${let}\:{I}\:=\:\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{8}}} \:\:{e}^{−\mathrm{2}{t}} \:{cos}^{\mathrm{4}} {t}\:\:\:\:{and}\:{J}\:\:=\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{8}}} \:{e}^{−\mathrm{2}{t}} \:{sin}^{\mathrm{4}} {dt} \\ $$$${find}\:{the}\:{values}\:{of}\:{I}\:{andJ}\:. \\ $$
Question Number 42798 Answers: 1 Comments: 2
$${find}\:{I}_{{n}} =\:\int_{\mathrm{0}} ^{\mathrm{1}} \:{x}^{{n}} \sqrt{\mathrm{1}−{x}^{\mathrm{2}} }{dx} \\ $$
Question Number 42797 Answers: 0 Comments: 1
$${let}\:{u}_{{k}} =\:\int_{−\frac{\pi}{\mathrm{2}}\:+{k}\pi} ^{−\frac{\pi}{\mathrm{2}}\:+\left({k}+\mathrm{1}\right)\pi} \:\:{e}^{−{t}} \:{cost}\:{dt} \\ $$$$\left.\mathrm{1}\right)\:{calculate}\:{u}_{{k}} \\ $$$$\left.\mathrm{2}\right)\:{let}\:{A}_{{n}} =\sum_{{k}=\mathrm{0}} ^{{n}} \:{u}_{{k}} \:\:\:\:\:{find}\:{lim}_{{n}\rightarrow+\infty} \:{A}_{{n}} \\ $$
Question Number 42796 Answers: 1 Comments: 1
$${calculate}\:{I}\:=\:\int_{\mathrm{0}} ^{\mathrm{1}} \:\:\frac{{x}^{\mathrm{2}} }{\mathrm{1}+{x}^{\mathrm{2}} }\:{arctan}\left({x}\right){dx} \\ $$
Question Number 42795 Answers: 0 Comments: 3
$${calculate}\:\:\int_{−\infty} ^{+\infty} \:\:\:\:\frac{{dx}}{\left(\mathrm{1}+{x}^{\mathrm{2}} \:+{x}^{\mathrm{4}} \right)^{\mathrm{2}} } \\ $$
Question Number 42793 Answers: 1 Comments: 0
$${find}\:{f}\left({a}\right)=\:\int_{\mathrm{0}} ^{\mathrm{1}} \:\:\:\frac{{dt}}{\left({a}^{\mathrm{2}} \:+{t}^{\mathrm{2}} \right)^{\mathrm{3}} }\:\:\:{with}\:{a}>\mathrm{0} \\ $$
Question Number 42792 Answers: 1 Comments: 1
$${find}\:\int_{\mathrm{0}} ^{\mathrm{1}} \:\:\frac{\mathrm{1}+{x}^{\mathrm{2}} }{\mathrm{1}+{x}^{\mathrm{3}} }{dx}\: \\ $$
Question Number 42791 Answers: 2 Comments: 0
$${find}\:\:\int\:\:\:\:\frac{{dx}}{\left({x}^{\mathrm{2}\:} +\mathrm{1}\right)\sqrt{\mathrm{1}+{x}^{\mathrm{2}} }} \\ $$
Question Number 42790 Answers: 1 Comments: 0
$$\left.\mathrm{1}\right)\:{calculate}\:\:\int_{\mathrm{0}} ^{\infty} \:\:\frac{{t}}{\mathrm{1}+{t}^{\mathrm{4}} }\:{dt} \\ $$$$\left.\mathrm{2}\right)\:{calculate}\:\int_{\mathrm{0}} ^{\mathrm{1}} \:\:\:\frac{{t}}{\mathrm{1}+{t}^{\mathrm{4}} }{dt} \\ $$$$\left.\mathrm{3}\right)\:{calculste}\:\int_{\mathrm{1}} ^{+\infty} \:\:\frac{{t}}{\mathrm{1}+{t}^{\mathrm{4}} }{dt}\: \\ $$
Question Number 42773 Answers: 0 Comments: 0
$${let}\:{f}\left({x}\right)\:=\:\int_{\mathrm{0}} ^{\mathrm{1}} \:\:\:\frac{{e}^{{t}} }{\mathrm{1}+{x}^{{t}} }\:{dt}\:\:\:\:\:{with}\:\mathrm{0}<{x}<\mathrm{1} \\ $$$${give}\:{f}\left({x}\right)\:{at}\:{form}\:{of}\:{serie}\:. \\ $$
Question Number 42771 Answers: 0 Comments: 0
$$\left.\mathrm{1}\right)\:{find}\:\int_{\mathrm{0}} ^{\mathrm{1}} \:\:\frac{{ln}\left({x}\right)}{\mathrm{1}−{x}^{\mathrm{2}} }{dx} \\ $$$$\left.\mathrm{2}\right)\:{find}\:\int_{\mathrm{0}} ^{\mathrm{1}} \:\frac{{ln}\left({x}\right)}{\mathrm{1}−{x}^{\mathrm{4}} }{dx} \\ $$
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