Question Number 68869 by mathmax by abdo last updated on 16/Sep/19 $${let}\:{f}\left({a}\right)\:=\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \:\:\frac{{dx}}{{a}+{sinx}}\:\:\:\:\:\left({a}\:{real}\right) \\ $$$$\left.\mathrm{1}\right){find}\:{a}\:{explicit}\:{form}\:\:{for}\:{f}\left({a}\right) \\ $$$$\left.\mathrm{2}\right)\:{calculste}\:{also}\:{g}\left({a}\right)=\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \:\:\frac{{dx}}{\left({a}+{sinx}\right)^{\mathrm{2}} }\:\:{and}\:{h}\left({a}\right)=\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \:\frac{{dx}}{\left({a}+{sinx}\right)^{\mathrm{3}} } \\…
Question Number 68788 by mhmd last updated on 15/Sep/19 $$\int\mathrm{1}/\left(\mathrm{1}+{x}^{\mathrm{2}} \right)^{{n}} \:{dx} \\ $$ Commented by Prithwish sen last updated on 15/Sep/19 $$\mathrm{Using}\:\mathrm{by}\:\mathrm{parts} \\ $$$$\mathrm{I}_{\mathrm{n}}…
Question Number 134301 by bramlexs22 last updated on 02/Mar/21 $$\Omega\:=\:\int_{\mathrm{0}} ^{\infty} \frac{{x}^{\mathrm{2}} }{\left(\mathrm{1}+{x}^{\mathrm{2}} \right)^{\mathrm{4}} }\:{dx} \\ $$ Answered by EDWIN88 last updated on 02/Mar/21 $$\mathrm{replace}\:\mathrm{x}\:\mathrm{by}\:\frac{\mathrm{1}}{\mathrm{x}}\:\mathrm{yields}\:…
Question Number 134303 by bramlexs22 last updated on 02/Mar/21 $$\mathcal{F}=\int_{\mathrm{0}} ^{\infty} \frac{\mathrm{16}\:\mathrm{arctan}\:\left({x}\right)}{\mathrm{1}+{x}^{\mathrm{2}} }\:{dx} \\ $$ Answered by Ñï= last updated on 02/Mar/21 $$\mathcal{F}=\int_{\mathrm{0}} ^{\infty} \frac{\mathrm{16}\:\mathrm{arctan}\:\left({x}\right)}{\mathrm{1}+{x}^{\mathrm{2}}…
Question Number 68767 by aliesam last updated on 15/Sep/19 Commented by ~ À ® @ 237 ~ last updated on 15/Sep/19 $${Let}\:{named}\:{it}\:\:{f}_{{n}} \left({x}\right) \\ $$$${Let}\:{named}\:\:\forall\:{a}>\mathrm{0}\:\:{g}\left({a},{x}\right)=\int\:\frac{{dx}}{{a}+{x}^{\mathrm{2}}…
Question Number 134302 by bramlexs22 last updated on 02/Mar/21 $$\int_{\mathrm{0}} ^{\:\mathrm{1}} \:\frac{\mathrm{ln}\:{x}}{\:\sqrt{\mathrm{1}−{x}^{\mathrm{2}} }}\:\mathrm{dx} \\ $$ Answered by Dwaipayan Shikari last updated on 02/Mar/21 $${I}\left({a}\right)=\int_{\mathrm{0}} ^{\mathrm{1}}…
Question Number 134289 by bramlexs22 last updated on 02/Mar/21 $$\mathrm{I}=\int\:\frac{\mathrm{x}^{\mathrm{n}} }{\:\sqrt{\mathrm{ax}+\mathrm{b}}}\:\mathrm{dx} \\ $$$$\mathrm{H}=\int\:\frac{\mathrm{x}^{\mathrm{4}} }{\:\sqrt{\mathrm{2x}+\mathrm{1}}}\:\mathrm{dx} \\ $$ Answered by EDWIN88 last updated on 02/Mar/21 $$\mathrm{I}_{\mathrm{n}} =\:\frac{\mathrm{2x}^{\mathrm{n}}…
Question Number 134291 by Lordose last updated on 02/Mar/21 $$ \\ $$$$\:\boldsymbol{\mathrm{Prove}}\:\:\:\int_{\mathrm{0}} ^{\:\infty} \frac{\mathrm{x}^{\mathrm{a}} }{\mathrm{1}+\mathrm{e}^{\mathrm{x}} }\mathrm{dx}\:=\:\left(\mathrm{1}−\mathrm{2}^{−\mathrm{a}} \right)\boldsymbol{\zeta}\left(\mathrm{a}+\mathrm{1}\right)\boldsymbol{\Gamma}\left(\mathrm{a}+\mathrm{1}\right) \\ $$$$ \\ $$ Answered by Dwaipayan Shikari…
Question Number 134272 by mnjuly1970 last updated on 01/Mar/21 $$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\mathrm{nice}\:\:\:\mathrm{calculus} \\ $$$$\:\:\:\:\mathrm{prove}\:\mathrm{that}:\::\: \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:{i}\:::\:\: =\int_{\mathrm{0}} ^{\:\infty} \frac{{cos}\left(\pi{x}\right)}{{e}^{\mathrm{2}\pi\sqrt{{x}}\:} −\mathrm{1}}\:{dx}=\frac{\mathrm{2}−\sqrt{\mathrm{2}}\:}{\mathrm{8}} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:{ii}::\:{compute}:\:\:\underset{{n}=−\infty} {\overset{\infty} {\sum}}\frac{\mathrm{1}}{{n}^{\mathrm{4}} +\mathrm{9}{n}^{\mathrm{2}} +\mathrm{10}}\:=? \\…
Question Number 3190 by Filup last updated on 07/Dec/15 $$\mathrm{How}\:\mathrm{do}\:\mathrm{you}\:\mathrm{find}\:\mathrm{the}\:\mathrm{length}\:\mathrm{of} \\ $$$$\mathrm{a}\:\mathrm{curve}\:\mathrm{between}\:{f}\left({a}\right)\:\mathrm{and}\:{f}\left({b}\right)? \\ $$ Commented by 123456 last updated on 07/Dec/15 $$\mathrm{wath}\:\mathrm{you}\:\mathrm{mean}\:\mathrm{by}\:\mathrm{lenght}\:\mathrm{of}\:\mathrm{curve}? \\ $$$$\mathrm{if}\:\mathrm{i}\:\mathrm{undestand}\:\mathrm{right},\:\mathrm{if}\:{f}\:\mathrm{is}\:\mathrm{continuous} \\…