Question Number 35611 by abdo mathsup 649 cc last updated on 21/May/18 $${let}\:{h}\left({t}\right)\:=\:{e}^{{t}−{e}^{{t}} } \:\:\:\:{and}\:{for}\:{n}\geqslant\mathrm{0}\:{we}\:{put} \\ $$$${h}_{{n}} \left({t}\right)\:={nh}\left({nt}\right) \\ $$$${calculate}\:\:\int_{−\infty} ^{+\infty} \:{h}_{{n}} \left({t}\right){dt}\:. \\ $$…
Question Number 35612 by abdo mathsup 649 cc last updated on 21/May/18 $${calculate}\:{I}\:=\int_{\mathrm{0}} ^{\infty} \:\:\:\frac{\left(\mathrm{1}+{t}\right)^{−\frac{\mathrm{1}}{\mathrm{4}}} \:\:\:−\left(\mathrm{1}+{t}\right)^{−\frac{\mathrm{3}}{\mathrm{4}}} }{{t}}{dt}\: \\ $$ Terms of Service Privacy Policy Contact:…
Question Number 35610 by abdo mathsup 649 cc last updated on 21/May/18 $$\left.{let}\:{give}\:{x}\in\right]\mathrm{0},\mathrm{2}\pi\left[\:\:{and}\:{a}\:\in{R},{b}\in\:{R}\right. \\ $$$${prove}\:{that}\:\:\frac{\pi−{x}}{\mathrm{2}}\:=\:{arctan}\left(\frac{{sinx}}{\mathrm{1}−{cosx}}\right) \\ $$$$\left.\mathrm{2}\right)\:{prove}\:{that}\:\mid{arctan}\left({a}\right)−{arctan}\left({b}\right)\mid\leqslant\mid{a}−{b}\mid \\ $$$$\left.\mathrm{3}\left.\right){let}\theta\:\in\right]\mathrm{0},\frac{\pi}{\mathrm{2}}\left[\:\:,\:{x}\:\in\left[\theta,\mathrm{2}\pi−\theta\right]\:,\:{r}\in\left[\mathrm{0},\mathrm{1}\left[\:{prove}\:{that}\right.\right.\right. \\ $$$$\mid\varphi\left({x},{r}\right)\:−\frac{\pi−{x}}{\mathrm{2}}\mid\leqslant\:\:\frac{\mathrm{1}−{r}}{\left(\mathrm{1}−{cos}\theta\right)^{\mathrm{2}} } \\ $$ Terms…
Question Number 35608 by abdo mathsup 649 cc last updated on 21/May/18 $${let}\:{r}\in\left[\mathrm{0},\mathrm{1}\left[\:{and}\:{x}\:{from}\:{R}\right.\right. \\ $$$${F}\left({x},{r}\right)\:=\:\frac{\mathrm{1}}{\mathrm{2}\pi}\:\int_{\mathrm{0}} ^{\mathrm{2}\pi} \:\:\:\:\frac{\left(\mathrm{1}−{r}^{\mathrm{2}} \right){f}\left({t}\right)}{\mathrm{1}−\mathrm{2}{r}\:{cos}\left({t}−{x}\right)\:+{r}^{\mathrm{2}} }{dt}\:\:{with} \\ $$$${f}\:\:\in\:{C}^{\mathrm{0}} \left({R}\right)\:\:\mathrm{2}\pi\:{periodic}\:\:{and}\:\:\mid\mid{f}\mid\mid={sup}_{{t}\in{R}} \mid{f}\left({t}\right)\mid \\ $$$$\:{prove}\:{that}\:{F}\left({x},{r}\right)=\:\frac{{a}_{\mathrm{0}}…
Question Number 35609 by abdo mathsup 649 cc last updated on 21/May/18 $${let}\:{r}\:\in\left[\mathrm{0},\mathrm{1}\left[\:{and}\:{x}\in\:{R}\:\:{and}\:\right.\right. \\ $$$$\varphi\left({x},{r}\right)\:=\:{arctan}\left(\:\frac{{rsinx}}{\mathrm{1}−{r}\:{cosx}}\right) \\ $$$$\left.\mathrm{1}\right)\:{prove}\:{that}\:\:\frac{\partial\varphi}{\partial{x}}\left({x},{r}\right)\:\:=\sum_{{n}=\mathrm{1}} ^{\infty} \:{r}^{{n}} \:{cos}\left({nx}\right) \\ $$$$\left.\mathrm{2}\right){prove}\:{that}\:\varphi\left({x},{r}\right)\:=\:\sum_{{n}=\mathrm{1}} ^{\infty} \:{r}^{{n}} \:\:\frac{{sin}\left({nx}\right)}{{n}}…
Question Number 35605 by abdo mathsup 649 cc last updated on 21/May/18 $${let}\:{r}\in\left[\mathrm{0},\mathrm{1}\left[\:{and}\:\theta\:\in\:{R},{x}\in\:{R}\:{prove}\:{that}\right.\right. \\ $$$$\left.\mathrm{1}\right)\:\mathrm{1}+\:\mathrm{2}\:\sum_{{n}=\mathrm{1}} ^{+\infty} \:{r}^{{n}} {cos}\theta\:=\:\frac{\mathrm{1}−{r}^{\mathrm{2}} }{\mathrm{1}−\mathrm{2}{r}\:{cos}\theta\:+{r}^{\mathrm{2}} } \\ $$$$\left.\mathrm{2}\right)\mathrm{1}\:=\frac{\mathrm{1}}{\mathrm{2}\pi}\:\int_{\mathrm{0}} ^{\mathrm{2}\pi} \:\:\:\:\:\:\frac{\left(\mathrm{1}−{r}^{\mathrm{2}} \right)}{\mathrm{1}−\mathrm{2}{rcos}\left({t}−{x}\right)\:+{r}^{\mathrm{2}}…
Question Number 35603 by abdo mathsup 649 cc last updated on 20/May/18 $${let}\:{x}\:\in\:{R}\:\:{and}\:\left\{{x}\right\}={x}\:−\left[{x}\right] \\ $$$${prove}\:{that}\:\:\int_{\mathrm{1}} ^{+\infty} \:\frac{\left\{{x}\right\}}{{x}^{\mathrm{2}} }\:{dx}\:{is}\:{convergent}\:{and}\:{find} \\ $$$${its}\:{value}\:. \\ $$ Terms of Service…
Question Number 35590 by abdo mathsup 649 cc last updated on 20/May/18 $${find}\:\:{J}\:\:=\:\int_{\mathrm{0}} ^{\mathrm{1}} \:\:{e}^{−{ax}} {ln}\left(\mathrm{1}+{e}^{−{bx}} \right){dx}\:{with}\:{a}>\mathrm{0}\:{and} \\ $$$${b}>\mathrm{0}\:. \\ $$ Terms of Service Privacy…
Question Number 166660 by mnjuly1970 last updated on 24/Feb/22 $$ \\ $$$$\:\:\:{calculate} \\ $$$$\:\:\:\:\Omega\:=\:\underset{{n}=\mathrm{0}} {\overset{\infty} {\sum}}\frac{\mathrm{1}}{\left(\mathrm{3}{n}\right)!}\:\:=\:? \\ $$$$\:\:\:\:\: \\ $$ Answered by amin96 last updated…
Question Number 35593 by abdo.msup.com last updated on 20/May/18 $${let}\:{f}\left({a}\right)=\int_{\mathrm{0}} ^{\infty} \:\:{e}^{−\left(\:{t}^{\mathrm{2}} \:+\frac{{a}}{{t}^{\mathrm{2}} }\right)} {dt}\:{witha}>\mathrm{0} \\ $$$${find}\:{the}\:{value}\:{of}\:{f}\left({a}\right). \\ $$ Terms of Service Privacy Policy Contact:…