Question Number 130214 by Lordose last updated on 23/Jan/21 $$\int\frac{\mathrm{u}^{\mathrm{2}} }{\left(\mathrm{1}+\mathrm{u}^{\mathrm{2}} \right)^{\mathrm{2}} }\mathrm{du} \\ $$ Answered by liberty last updated on 23/Jan/21 $$\mathrm{J}=\int\:\frac{\left(\mathrm{u}^{\mathrm{2}} +\mathrm{1}\right)−\mathrm{1}}{\left(\mathrm{1}+\mathrm{u}^{\mathrm{2}} \right)^{\mathrm{2}}…
Question Number 130208 by liberty last updated on 23/Jan/21 $$\:\mathrm{The}\:\mathrm{loop}\:\mathrm{of}\:\mathrm{curve}\:\mathrm{2ay}^{\mathrm{2}} =\mathrm{x}\left(\mathrm{x}−\mathrm{a}\right)^{\mathrm{2}} \\ $$$$\mathrm{revolves}\:\mathrm{about}\:\mathrm{straight}\:\mathrm{line}\: \\ $$$$\mathrm{y}=\mathrm{a}.\:\mathrm{Find}\:\mathrm{the}\:\mathrm{volume}\:\mathrm{of}\:\mathrm{the}\:\mathrm{solid} \\ $$$$\mathrm{generated}. \\ $$ Answered by benjo_mathlover last updated on…
Question Number 130203 by mnjuly1970 last updated on 23/Jan/21 Answered by Dwaipayan Shikari last updated on 23/Jan/21 $${I}\left({a}\right)−{I}\left(\beta\right)=\int_{\mathrm{0}} ^{\mathrm{1}} \frac{{x}^{{a}} }{\left(\mathrm{1}+{x}\right){logx}}−\frac{{x}^{\beta} }{\left(\mathrm{1}+{x}\right){log}\left({x}\right)}{dx} \\ $$$${I}'\left({a}\right)−{I}'\left(\beta\right)=\underset{{n}=\mathrm{1}} {\overset{\infty}…
Question Number 64662 by aliesam last updated on 20/Jul/19 $$\int\frac{{dx}}{\left({x}^{\mathrm{8}} +{x}^{\mathrm{4}} +\mathrm{1}\right)^{\mathrm{2}} } \\ $$$$ \\ $$$$\int_{\frac{\mathrm{1}}{{x}}} ^{{x}} \frac{{ln}\left({t}\right)}{{t}^{\mathrm{2}} +\mathrm{1}}\:{dt} \\ $$ Commented by MJS…
Question Number 64649 by mathmax by abdo last updated on 20/Jul/19 $${calculate}\:\int_{\mathrm{0}} ^{\mathrm{2}\pi} \:\:\frac{{cos}\theta}{\mathrm{5}+\mathrm{3}{cos}\theta}{d}\theta \\ $$ Commented by mathmax by abdo last updated on 20/Jul/19…
Question Number 64642 by mmkkmm000m last updated on 19/Jul/19 $$\int\left({dx}\right)/{e}^{{x}} +{x} \\ $$ Commented by mathmax by abdo last updated on 20/Jul/19 $${let}\:{I}\:=\int\:\:\:\frac{{dx}}{{x}+{e}^{{x}} }\:\Rightarrow{I}\:=\int\:\:\:\frac{{e}^{−{x}} }{{xe}^{−{x}}…
Question Number 64635 by mathmax by abdo last updated on 19/Jul/19 $$\left.\mathrm{1}\right){calculate}\:{f}\left({a}\right)\:=\int_{\mathrm{0}} ^{\infty} \:\frac{{arctan}\left(\alpha{x}\right)}{\mathrm{1}+{x}^{\mathrm{2}} }{dx}\:\:{with}\:\alpha\:{real} \\ $$$$\left.\mathrm{2}\right)\:{find}\:{the}\:{value}\:{of}\:\int_{\mathrm{0}} ^{\infty} \:\:\frac{{arctan}\left(\mathrm{2}{x}\right)}{\mathrm{1}+{x}^{\mathrm{2}} }{dx} \\ $$ Commented by mathmax…
Question Number 130167 by Lordose last updated on 23/Jan/21 $$\int_{\mathrm{0}} ^{\:\infty} \frac{\mathrm{ln}\left(\mathrm{u}\right)\mathrm{e}^{−\mathrm{u}} }{\left(\mathrm{1}+\mathrm{e}^{−\mathrm{u}} \right)^{\mathrm{2}} }\mathrm{du} \\ $$$$ \\ $$$$ \\ $$ Answered by mindispower last…
Question Number 130158 by benjo_mathlover last updated on 23/Jan/21 $$\mathrm{Find}\:\mathrm{the}\:\mathrm{area}\:\mathrm{of}\:\mathrm{the}\:\mathrm{segment}\:\mathrm{of}\:\mathrm{the}\:\mathrm{curve} \\ $$$$\mathrm{y}^{\mathrm{2}} \:=\:\:\:\mathrm{x}^{\mathrm{3}} −\mathrm{x}^{\mathrm{2}} \:\mathrm{if}\:\mathrm{the}\:\mathrm{line}\:\mathrm{x}=\:\:\:\mathrm{2}\:\mathrm{is}\:\mathrm{the}\:\mathrm{chord}\: \\ $$$$\mathrm{determining}\:\mathrm{the}\:\mathrm{segment}\: \\ $$ Answered by liberty last updated on…
Question Number 130149 by liberty last updated on 22/Jan/21 $$\:\Re\:=\:\int\:\sqrt{{x}+\sqrt{{x}+\sqrt{{x}+\sqrt{{x}+…}}}}\:{dx} \\ $$$$\:\Im\:=\:\int_{\mathrm{0}} ^{\:\mathrm{1}} \:\frac{\ell\mathrm{n}\:{x}}{{x}+\mathrm{1}}\:{dx}\: \\ $$ Answered by MJS_new last updated on 22/Jan/21 $$\sqrt{{x}+\sqrt{{x}+\sqrt{{x}+…}}}=\frac{\mathrm{1}}{\mathrm{2}}+\frac{\sqrt{\mathrm{4}{x}+\mathrm{1}}}{\mathrm{2}} \\…