Question Number 45043 by maxmathsup by imad last updated on 07/Oct/18 $${let}\:{f}\left({x}\right)=\frac{\mathrm{1}+\sqrt{\mathrm{1}+{x}^{\mathrm{2}} }}{{x}} \\ $$$$\left.\mathrm{1}\right)\:{calculate}\:\:\int_{\mathrm{1}} ^{\mathrm{3}} {f}\left({x}\right){dx} \\ $$$$\left.\mathrm{2}\right)\:{determine}\:{f}^{−\mathrm{1}} \left({x}\right) \\ $$$$\left.\mathrm{3}\right)\:{find}\:\int\:{f}^{−\mathrm{1}} \left({x}\right){dx}\:\:. \\ $$…
Question Number 45044 by maxmathsup by imad last updated on 07/Oct/18 $${let}\:{f}\left({x}\right)\:={x}^{\mathrm{2}} \:,\:{function}\:\mathrm{2}\pi\:{peridic}\:{even} \\ $$$$\left.\mathrm{1}\right)\:{developp}\:{f}\:{at}\:{fourier}\:{serie} \\ $$$$\left.\mathrm{2}\right){find}\:{the}\:{value}\:{of}\:\sum_{{n}=\mathrm{1}} ^{\infty} \:\frac{\mathrm{1}}{{n}^{\mathrm{4}} } \\ $$ Commented by maxmathsup…
Question Number 45021 by rahul 19 last updated on 07/Oct/18 $$\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \frac{{dx}}{\mathrm{1}+\mathrm{tan}\:^{\mathrm{13}} {x}}\:=\:? \\ $$ Answered by tanmay.chaudhury50@gmail.com last updated on 07/Oct/18 $${I}=\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}}…
Question Number 45019 by rahul 19 last updated on 07/Oct/18 Answered by tanmay.chaudhury50@gmail.com last updated on 08/Oct/18 $${let}\:{t}=\mathrm{1}+\frac{{x}}{\mathrm{1}!}+\frac{{x}^{\mathrm{2}} }{\mathrm{2}\centerdot!}+\frac{{x}^{\mathrm{3}} }{\mathrm{3}!}+…+\frac{{x}^{{n}} }{{n}!} \\ $$$$\frac{{dt}}{{dx}}=\mathrm{0}+\mathrm{1}+\frac{{x}}{\mathrm{1}!}+\frac{{x}^{\mathrm{2}} }{\mathrm{2}!}+..+\frac{{x}^{{n}−\mathrm{1}} }{\left({n}−\mathrm{1}\right)!}+\frac{{x}^{{n}}…
Question Number 45020 by rahul 19 last updated on 07/Oct/18 Commented by maxmathsup by imad last updated on 07/Oct/18 $${let}\:{decompose}\:{F}\left({x}\right)\:=\frac{\mathrm{1}}{{x}^{\mathrm{3}} \:+\mathrm{1}}\:\Rightarrow{F}\left({x}\right)=\frac{\mathrm{1}}{\left({x}+\mathrm{1}\right)\left({x}^{\mathrm{2}} −{x}+\mathrm{1}\right)} \\ $$$$=\frac{{a}}{{x}+\mathrm{1}}\:+\frac{{bx}+{c}}{{x}^{\mathrm{2}} −{x}\:+\mathrm{1}}…
Question Number 110551 by shahria14 last updated on 29/Aug/20 Answered by Dwaipayan Shikari last updated on 29/Aug/20 $$\int_{\mathrm{0}} ^{\pi} \frac{{x}}{\mathrm{1}+{sinx}}{dx}=\int_{\mathrm{0}} ^{\pi} \frac{\pi−{x}}{\mathrm{1}+{sinx}}{dx}={I} \\ $$$$\mathrm{2}{I}=\int_{\mathrm{0}} ^{\pi}…
Question Number 110549 by shahria14 last updated on 29/Aug/20 Commented by peter frank last updated on 29/Aug/20 Answered by peter frank last updated on 29/Aug/20…
Question Number 110543 by mnjuly1970 last updated on 29/Aug/20 Answered by mathdave last updated on 29/Aug/20 $${solution} \\ $$$${let}\:{y}={x}^{\mathrm{2}} \:\:{and}\:{dx}=\frac{\mathrm{1}}{\mathrm{2}\sqrt{{y}}}{dy}\:\:{then}\:{putting}\:{into} \\ $$$${I}=\frac{\mathrm{1}}{\mathrm{2}}\int_{\mathrm{0}} ^{\mathrm{1}} \frac{\mathrm{ln}\left(\mathrm{1}−{y}\right)\mathrm{ln}\left(\mathrm{1}+{y}\right)}{{y}}{dy} \\…
Question Number 44994 by manish09@gmail.com last updated on 07/Oct/18 $$\int\mathrm{e}^{\mathrm{x}} \left(\frac{\mathrm{1}+\mathrm{sin}\:\mathrm{x}}{\mathrm{1}−\mathrm{cos}\:\mathrm{x}}\right)\mathrm{dx}\:=\:? \\ $$ Answered by arvinddayama01@gmail.com last updated on 07/Oct/18 $$\mathrm{i}\:\mathrm{think}\:\mathrm{problem}\:\mathrm{should}\:\mathrm{be}:− \\ $$$$\:\:\:\:\:\:\int\mathrm{e}^{\mathrm{x}} \left(\frac{\mathrm{1}−\mathrm{sin}\:\mathrm{x}}{\mathrm{1}−\mathrm{cosx}}\right)\mathrm{dx} \\…
Question Number 44992 by manish09@gmail.com last updated on 07/Oct/18 $$\int\mathrm{e}^{\mathrm{x}} \left(\frac{\mathrm{1}−\mathrm{sin}\:\mathrm{x}}{\mathrm{1}+\mathrm{cos}\:\mathrm{x}}\right)\mathrm{dx}\:? \\ $$ Answered by tanmay.chaudhury50@gmail.com last updated on 07/Oct/18 $$\int{e}^{{x}} \left(\frac{\mathrm{1}−\mathrm{2}{sin}\frac{{x}}{\mathrm{2}}{cos}\frac{{x}}{\mathrm{2}}}{\mathrm{2}{cos}^{\mathrm{2}} \frac{{x}}{\mathrm{2}}}\right) \\ $$$$\int{e}^{{x}}…