Question Number 118260 by bobhans last updated on 16/Oct/20 $$\int\:\frac{{x}^{\mathrm{2}} −\mathrm{1}}{{x}^{\mathrm{4}} +{x}^{\mathrm{2}} +\mathrm{1}}\:{dx} \\ $$ Commented by bobhans last updated on 16/Oct/20 $${yes}…….\:{all}\:{sir} \\ $$…
Question Number 183794 by cortano1 last updated on 30/Dec/22 $$\:{If}\:\underset{\mathrm{0}} {\overset{\pi} {\int}}\:\frac{\mathrm{cos}\:{x}}{\left({x}+\mathrm{2}\right)^{\mathrm{2}} }\:{dx}=\:{T} \\ $$$$\:{then}\:\underset{\mathrm{0}} {\overset{\pi} {\int}}\:\frac{\mathrm{sin}\:\mathrm{2}{x}}{{x}+\mathrm{1}}\:{dx}\:=\:?\: \\ $$ Commented by Frix last updated on…
Question Number 118246 by bemath last updated on 16/Oct/20 $$\:\:\:\:\underset{\mathrm{0}} {\overset{\mathrm{1}} {\int}}\:\frac{\mathrm{ln}\:{x}}{{x}+\mathrm{1}}\:{dx}\:=? \\ $$ Answered by Dwaipayan Shikari last updated on 16/Oct/20 $$\int_{\mathrm{0}} ^{\mathrm{1}} \frac{{logx}}{{x}+\mathrm{1}}{dx}=\left[{log}\left({x}\right){log}\left({x}+\mathrm{1}\right)\right]_{\mathrm{0}}…
Question Number 52703 by maxmathsup by imad last updated on 11/Jan/19 $${let}\:{f}\left({t}\right)\:=\int_{\mathrm{0}} ^{\infty} \:\:\frac{{cos}^{\mathrm{2}} \left({tx}\right)}{\left({x}^{\mathrm{2}} +\mathrm{3}\right)^{\mathrm{2}} }\:{dx}\:\:{with}\:{t}\:\geqslant\mathrm{0} \\ $$$$\left.\mathrm{1}\right)\:{give}\:{a}\:{explicit}\:{form}\:{of}\:{f}\left({t}\right) \\ $$$$\left.\mathrm{2}\right)\:{find}\:{the}\:{value}\:{of}\:\:\int_{\mathrm{0}} ^{\infty} \:\:\frac{{xsin}\left(\mathrm{2}{tx}\right)}{\left({x}^{\mathrm{2}} \:+\mathrm{3}\right)^{\mathrm{2}} }\:{dx}…
Question Number 118230 by bobhans last updated on 16/Oct/20 $$\mathrm{If}\:\int\:\frac{\left(\sqrt{\mathrm{x}}\right)^{\mathrm{5}} }{\left(\sqrt{\mathrm{x}}\right)^{\mathrm{7}} +\mathrm{x}^{\mathrm{6}} }\:\mathrm{dx}\:=\:\mathrm{p}\:\mathrm{ln}\:\left(\frac{\mathrm{x}^{\mathrm{q}} }{\mathrm{x}^{\mathrm{q}} +\mathrm{1}}\right)\:+\:\mathrm{C}\: \\ $$$$\mathrm{find}\:\mathrm{the}\:\mathrm{value}\:\mathrm{of}\:\mathrm{p}\:\mathrm{and}\:\mathrm{q}. \\ $$ Answered by bemath last updated on…
Question Number 183761 by MikeH last updated on 29/Dec/22 $$\mathrm{Solve}\:\mathrm{the}\:\mathrm{differential}\:\mathrm{equation}\:\mathrm{for}\:\mathrm{the}\:\mathrm{function} \\ $$$$\mathrm{given}\:\mathrm{by}\:{U}\left({x},{t}\right). \\ $$$$\begin{cases}{\frac{\partial{U}}{\partial{t}}\:=\:\mathrm{2}\frac{\partial^{\mathrm{2}} {U}}{\partial{x}^{\mathrm{2}} }\:,\:\mathrm{0}\:<\:{x}\:<\:\pi}\\{{U}\left(\mathrm{0},{t}\right)\:=\:\mathrm{0},\:{U}\left(\pi,{t}\right)\:=\:\mathrm{0},\:{t}\:>\:\mathrm{0}}\end{cases} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:{U}\left({x},\mathrm{0}\right)\:=\:\mathrm{25}{x} \\ $$ Answered by leodera last updated…
Question Number 118218 by bemath last updated on 16/Oct/20 $$\:\int\:\mathrm{sin}\:^{\mathrm{6}} \left(\mathrm{2}{x}\right){dx}\:=?\: \\ $$ Answered by bobhans last updated on 16/Oct/20 $$\:\mathrm{Solve}\:\int\:\mathrm{sin}\:^{\mathrm{6}} \left(\mathrm{2x}\right)\:\mathrm{dx}\:. \\ $$$$\mathrm{by}\:\mathrm{De}'\mathrm{moivre}\:\mathrm{theorem}\: \\…
Question Number 52683 by maxmathsup by imad last updated on 11/Jan/19 $${let}\:{f}\left(\lambda\right)\:=\int_{−\infty} ^{+\infty} \:\:\frac{{sin}\left(\lambda{x}\right)}{\left({x}^{\mathrm{2}} \:+\mathrm{2}\lambda{x}\:+\mathrm{1}\right)^{\mathrm{2}} }{dx}\:\:\:\:{with}\:\mid\lambda\mid<\mathrm{1} \\ $$$$\left.\mathrm{1}\right)\:{find}\:{the}\:{value}\:{of}\:{f}\left(\lambda\right) \\ $$$$\left.\mathrm{2}\right)\:{calculate}\:\int_{−\infty} ^{+\infty} \:\:\:\frac{{sin}\left(\frac{{x}}{\mathrm{2}\:}\right)}{\left({x}^{\mathrm{2}} \:\:+{x}+\mathrm{1}\right)^{\mathrm{2}} }{dx} \\…
Question Number 52680 by maxmathsup by imad last updated on 11/Jan/19 $${let}\:{f}_{{n}} \left({x}\right)=\frac{{sin}\left({nx}\right)}{{n}^{\mathrm{3}} }\:\:\:{and}\:{f}\left({x}\right)=\sum_{{n}=\mathrm{1}} ^{\infty} \:{f}_{{n}} \left({x}\right) \\ $$$${calculate}\:\int_{\mathrm{0}} ^{\pi} \:{f}\left({x}\right){dx}\:. \\ $$ Commented by…
Question Number 52667 by gunawan last updated on 11/Jan/19 $$\int\frac{{x}^{\mathrm{4}} +\mathrm{1}}{{x}^{\mathrm{2}} \sqrt{{x}^{\mathrm{4}} −\mathrm{1}}}\:{dx} \\ $$ Answered by tanmay.chaudhury50@gmail.com last updated on 11/Jan/19 $$\int\frac{{x}^{\mathrm{2}} +\frac{\mathrm{1}}{{x}^{\mathrm{2}} }}{\:\sqrt{{x}^{\mathrm{2}}…