Question Number 63665 by mathmax by abdo last updated on 07/Jul/19 $${find}\:{the}\:{value}\:{of}\:\sum_{{n}=\mathrm{1}} ^{\infty} \:\:\frac{\left(−\mathrm{1}\right)^{{n}} }{{n}^{\mathrm{2}} \left({n}+\mathrm{1}\right)^{\mathrm{3}} } \\ $$ Commented by mathmax by abdo last…
Question Number 63664 by mathmax by abdo last updated on 07/Jul/19 $${let}\:{f}\left({x}\right)=\int_{\mathrm{0}} ^{\infty} \:\:\frac{{t}^{{a}−\mathrm{1}} }{{x}+{t}^{{n}} }\:{dt}\:\:\:{with}\:\mathrm{0}<{a}<\mathrm{1}\:\:{and}\:\:{x}>\mathrm{0}\:{and}\:{n}\geqslant\mathrm{2} \\ $$$$\left.\mathrm{1}\right)\:{determine}\:{a}\:{explicit}\:{form}\:{of}\:{f}\left({x}\right) \\ $$$$\left.\mathrm{2}\right)\:{calculate}\:{g}\left({x}\right)\:=\int_{\mathrm{0}} ^{\infty} \:\:\frac{{t}^{{a}−\mathrm{1}} }{\left({x}+{t}^{{n}} \right)^{\mathrm{2}} }\:{dt}…
Question Number 63663 by Tawa1 last updated on 07/Jul/19 Answered by mr W last updated on 07/Jul/19 $$\angle{ADB}=\mathrm{360}−\left(\mathrm{180}−\mathrm{6}−\mathrm{30}\right)−\left(\mathrm{180}−\mathrm{6}−\mathrm{24}\right)=\mathrm{66}° \\ $$$$\frac{{AD}}{{DC}}=\frac{\mathrm{sin}\:\mathrm{30}°}{\mathrm{sin}\:\mathrm{6}°} \\ $$$$\frac{{BD}}{{DC}}=\frac{\mathrm{sin}\:\mathrm{24}°}{\mathrm{sin}\:\mathrm{6}°} \\ $$$$\Rightarrow\frac{{AD}}{{BD}}=\frac{\mathrm{sin}\:\mathrm{30}°}{\mathrm{sin}\:\mathrm{24}°} \\…
Question Number 129199 by Adel last updated on 13/Jan/21 $$\underset{{x}\rightarrow\infty} {\mathrm{lim}}\left[\frac{\mathrm{x}^{\mathrm{x}+\mathrm{1}} }{\left(\mathrm{x}+\mathrm{1}\right)^{\mathrm{x}} }−\frac{\left(\mathrm{x}−\mathrm{1}\right)^{\mathrm{x}} }{\mathrm{x}^{\mathrm{x}−\mathrm{1}} }\right]=? \\ $$$$\mathrm{solve}\:\:\:\mathrm{tish}\:\:\mathrm{pleas} \\ $$ Answered by mr W last updated…
Question Number 63662 by mathmax by abdo last updated on 06/Jul/19 $$\:{let}\:{A}_{{n}} =\int_{\mathrm{0}} ^{\infty} \:\:\:\frac{{x}^{{a}−\mathrm{1}} }{\mathrm{1}+{x}^{{n}} }{dx}\:\:{with}\:{n}\:{integr}\:{and}\:{n}\geqslant\mathrm{2}\:\:{and}\:\mathrm{0}<{a}<\mathrm{1} \\ $$$$\left.\mathrm{1}\right)\:{calculate}\:{A}_{{n}} \\ $$$$\left.\mathrm{2}\right)\:{find}\:{the}\:{values}\:{of}\:\int_{\mathrm{0}} ^{\infty} \:\:\frac{{x}^{{a}−\mathrm{1}} }{\mathrm{1}+{x}^{\mathrm{2}} }{dx}\:{and}\:\int_{\mathrm{0}}…
Question Number 63661 by mathmax by abdo last updated on 06/Jul/19 $${let}\:\mathrm{0}<{a}<\mathrm{1}\:{find}\:{the}\:{valueof}\:\int_{\mathrm{0}} ^{\infty} \:\:\frac{{t}^{{a}−\mathrm{1}} }{\mathrm{1}+{t}^{\mathrm{2}} }{dt} \\ $$ Commented by mathmax by abdo last updated…
Question Number 63651 by mathmax by abdo last updated on 06/Jul/19 $${let}\:{S}_{{n}} \left({x}\right)=\sum_{{k}=\mathrm{0}} ^{{n}} \:{e}^{−{k}} {sin}\left({k}^{\mathrm{2}} {x}\right) \\ $$$$\left.\mathrm{1}\right)\:{determine}\:\mathrm{2}\:{sequence}\:\:{U}_{{n}} \left({x}\right)\:{and}\:{V}_{{n}} \left({x}\right)\:{wich}\:{verify}\:{U}_{{n}} \leqslant\:{S}_{{n}} \leqslant\:{V}_{{n}} \\ $$$$\left.\mathrm{2}\right)\:{let}\:\:{S}\:={lim}_{{n}\rightarrow+\infty}…
Question Number 129187 by math178 last updated on 13/Jan/21 Commented by math178 last updated on 13/Jan/21 $${differantial} \\ $$$${what}\:{is}\:{the}\:{special}\:{solution}? \\ $$$${thank}\:{you}\:<\mathrm{3} \\ $$ Answered by…
Question Number 129185 by Dwaipayan Shikari last updated on 13/Jan/21 $$\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \frac{{dx}}{\:\sqrt{\mathrm{97}+{sin}^{\mathrm{2}} {x}}}\:\:{Hypergeometric}\:{form} \\ $$ Commented by Dwaipayan Shikari last updated on 13/Jan/21 $${I}\:{have}\:{found}\:\frac{\pi}{\mathrm{2}\sqrt{\mathrm{97}}}\:\:_{\mathrm{2}}…
Question Number 129180 by mnjuly1970 last updated on 13/Jan/21 $$\:\:\:\:\:\:\:\:\:\:\:\:…\:{nice}\:\:\:{calculus}… \\ $$$$\:\:\:{calculate}:: \\ $$$$\:\:\:\:\:\:\phi\overset{???} {=}\int_{\mathrm{0}} ^{\:\frac{\pi}{\mathrm{4}}} \frac{{dx}}{\left({sin}\left({x}\right)+{cos}\left({x}\right)+\sqrt{\mathrm{2}}\:\right)^{\mathrm{2}} } \\ $$$$ \\ $$ Commented by Dwaipayan…