Category: Integration

Question Number 86830 by Ar Brandon last updated on 31/Mar/20 $${Prove}\:\:{that}\:\int_{\mathrm{0}} ^{\infty} \frac{{sin}\:{x}}{{x}}{dx}\:=\:\frac{\pi}{\mathrm{2}} \\ $$ Answered by mind is power last updated on 31/Mar/20 $${let}\:{f}\left({z}\right)=\frac{{e}^{{iz}}…

Question Number 152345 by mnjuly1970 last updated on 27/Aug/21 $$ \\ $$$$\:\:\:\:{prove}… \\ $$$$\:\:\:\:\:\mathrm{S}\:=\underset{{n}=\mathrm{1}} {\overset{\infty} {\sum}}\frac{\mathrm{1}}{}\:=\:\frac{\mathrm{1}}{\mathrm{3}}\:+\frac{\mathrm{2}\pi\:\sqrt{\mathrm{3}}}{\mathrm{27}}\:…\blacksquare \\ $$$$ \\ $$ Answered by Kamel last updated…

Question Number 152291 by iloveisrael last updated on 27/Aug/21 $$\:\int\:\frac{{dx}}{\mathrm{cos}\:{x}+\mathrm{cosec}\:{x}}\:=? \\ $$ Answered by puissant last updated on 27/Aug/21 $${I}=\int\frac{{dx}}{{cosx}+{cosecx}} \\ $$$$=\int\frac{{sinx}}{{cosxsinx}+\mathrm{1}}{dx}\:=\:\int\frac{\mathrm{2}{sinx}}{{sin}\mathrm{2}{x}+\mathrm{2}}{dx} \\ $$$$=\int\frac{{sinx}+{cosx}}{{sin}\mathrm{2}{x}+\mathrm{2}}{dx}+\int\frac{{sinx}−{cosx}}{{sin}\mathrm{2}{x}+\mathrm{2}}{dx} \\…

Question Number 152276 by peter frank last updated on 27/Aug/21 $$\mathrm{Prove}\:\mathrm{that} \\ $$$$\int_{\mathrm{0}} ^{\pi} \frac{\mathrm{xtan}\:\mathrm{x}}{\mathrm{sec}\:\mathrm{x}+\mathrm{tan}\:\mathrm{x}}\mathrm{dx}=\frac{\pi^{\mathrm{2}} }{\mathrm{2}}−\pi \\ $$ Answered by Olaf_Thorendsen last updated on 27/Aug/21…

Question Number 152275 by peter frank last updated on 27/Aug/21 $$\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \mathrm{sin}\:\mathrm{2xlog}\left(\:\mathrm{tan}\:\mathrm{x}\right)\mathrm{dx} \\ $$ Answered by qaz last updated on 27/Aug/21 $$\int_{\mathrm{0}} ^{\pi/\mathrm{2}} \mathrm{sin}\:\mathrm{2x}\centerdot\mathrm{lntan}\:\mathrm{xdx}…