Question Number 70651 by sadimuhmud 136 last updated on 06/Oct/19 Commented by Prithwish sen last updated on 06/Oct/19 $$\mathrm{Let}\:\sqrt{\mathrm{x}}\:=\:\mathrm{u}\:\:\Rightarrow\:\mathrm{d}\left(\sqrt{\mathrm{x}}\right)=\mathrm{du} \\ $$$$\int_{\mathrm{0}} ^{\frac{\mathrm{2}}{\:\sqrt{\mathrm{a}}}} \mathrm{e}^{\sqrt{\mathrm{a}}\mathrm{u}} \mathrm{du}\:\:=\:\frac{\mathrm{1}}{\:\sqrt{\mathrm{a}}}\:\left[\boldsymbol{\mathrm{e}}^{\sqrt{\boldsymbol{\mathrm{a}}}\boldsymbol{\mathrm{u}}} \right]_{\mathrm{0}}…
Question Number 136170 by mnjuly1970 last updated on 19/Mar/21 $$\:\:\:\:\:\:\:\:\:\:\:\:…..{nice}\:\:{calculus}…. \\ $$$$\:\:\:{compute}:: \\ $$$$\mathrm{2}{li}_{\mathrm{2}} \left(\frac{−\mathrm{1}}{\mathrm{2}}\right)−\mathrm{2}{li}_{\mathrm{2}} \left(\frac{\mathrm{1}}{\mathrm{2}}\right)+{li}_{\mathrm{2}} \left(\frac{\mathrm{3}}{\mathrm{4}}\right)=?? \\ $$$$ \\ $$ Answered by mindispower last…
Question Number 70602 by oyemi kemewari last updated on 06/Oct/19 $$\mathrm{I}=\int_{\mathrm{0}} ^{\infty} \mathrm{ln}\left(\sqrt{\mathrm{1}−\mathrm{x}}\:+\sqrt{\mathrm{1}+\mathrm{x}}\right)\mathrm{dx} \\ $$ Commented by mathmax by abdo last updated on 06/Oct/19 $${error}\:{in}\:{the}\:{question}\:{the}\:{function}\:{x}\rightarrow\sqrt{\mathrm{1}−{x}}+\sqrt{\mathrm{1}+{x}}\:{is}\:{defined}\:{on}…
Question Number 5065 by gourav~ last updated on 07/Apr/16 $$\int\sqrt{\mathrm{16}{x}^{\mathrm{2}} +\mathrm{25}}\:\:{dx}\:=? \\ $$ Answered by Yozzii last updated on 08/Apr/16 $${Let}\:{x}=\frac{\mathrm{5}}{\mathrm{4}}{sinh}\theta\Rightarrow{dx}=\frac{\mathrm{5}}{\mathrm{4}}{cosh}\theta{d}\theta \\ $$$$\sqrt{\mathrm{16}{x}^{\mathrm{2}} +\mathrm{25}}=\sqrt{\mathrm{16}×\frac{\mathrm{25}}{\mathrm{16}}{sinh}^{\mathrm{2}} \theta+\mathrm{25}}…
Question Number 136132 by frc2crc last updated on 19/Mar/21 $${Find}\:{a}\:{series}\:{for}\:\frac{{x}^{\mathrm{2}} }{\mathrm{tanh}\:\left({x}\pi\right)\mathrm{tan}\:\left({x}\pi\right)} \\ $$ Answered by Dwaipayan Shikari last updated on 19/Mar/21 $${sinh}\left(\pi{x}\right)=\pi{x}\underset{{n}=\mathrm{1}} {\overset{\infty} {\prod}}\left(\mathrm{1}+\frac{{x}^{\mathrm{2}} }{{n}^{\mathrm{2}}…
Question Number 70594 by mathmax by abdo last updated on 06/Oct/19 $${calculate}\:{by}\:{residus}\:{method}\:{the}\:{integral}\:\:\int_{\mathrm{0}} ^{\infty} \:\:\frac{{dx}}{\left(\mathrm{1}+{x}^{\mathrm{2}} \right)^{{n}} } \\ $$$${with}\:{n}\:{integr}\:{and}\:{n}\geqslant\mathrm{1} \\ $$ Commented by mathmax by abdo…
Question Number 5058 by Yozzii last updated on 06/Apr/16 $$\int\int\frac{{a}}{\left(\mathrm{1}−{lnx}\right)\left({lny}−\mathrm{1}\right)}{dxdy}=? \\ $$ Commented by Yozzii last updated on 06/Apr/16 $$\int\int\int\int…\int\int\int\frac{{a}}{\underset{{i}=\mathrm{1}} {\overset{{n}} {\prod}}\left(\mathrm{1}−{lnx}_{{i}} \right)}{dx}_{\mathrm{1}} {dx}_{\mathrm{2}} …{dx}_{{n}}…
Question Number 70571 by ahmadshahhimat775@gmail.com last updated on 05/Oct/19 Commented by mathmax by abdo last updated on 05/Oct/19 $$\left.\exists\:{c}\:\in\right]\mathrm{3},\mathrm{3}+{h}\left[\:/\int_{\mathrm{3}} ^{{h}+\mathrm{3}} \:\frac{\mathrm{5}{dx}}{{x}^{\mathrm{3}} \:+\mathrm{7}}\:=\frac{\mathrm{5}}{{c}^{\mathrm{3}} \:+\mathrm{7}}\:\int_{\mathrm{3}} ^{{h}+\mathrm{3}} {dx}=\frac{\mathrm{5}{h}}{{c}^{\mathrm{3}}…
Question Number 5032 by gourav~ last updated on 04/Apr/16 $$\int\frac{{ax}+{b}}{\left({cx}+{d}\right)^{\mathrm{2}} }{dx}\:=? \\ $$$$ \\ $$ Answered by Yozzii last updated on 04/Apr/16 $${Let}\:{u}={cx}+{d}\Rightarrow{du}={cdx}\Rightarrow{c}^{−\mathrm{1}} {du}={dx}. \\…
Question Number 5026 by gourav~ last updated on 04/Apr/16 $$\int\frac{\mathrm{16}}{{x}^{\mathrm{2}} \sqrt{\mathrm{4}−\mathrm{9}{x}^{\mathrm{2}} }}{dx}\:=? \\ $$$$ \\ $$ Answered by Yozzii last updated on 04/Apr/16 $${Let}\:{x}=\frac{\mathrm{2}}{\mathrm{3}}{cosy}\Rightarrow{dx}=−\frac{\mathrm{2}}{\mathrm{3}}{sinydy} \\…