Question Number 48719 by Abdo msup. last updated on 27/Nov/18 $${find}\:\:\int\:\:\:\:\frac{{x}−\mathrm{2}}{\:\sqrt{{x}^{\mathrm{2}} +\mathrm{4}{x}−\mathrm{3}}}{dx} \\ $$ Commented by Abdo msup. last updated on 27/Nov/18 $${I}=\int\:\:\frac{{x}−\mathrm{2}}{\:\sqrt{{x}^{\mathrm{2}} \:+\mathrm{4}{x}+\mathrm{4}−\mathrm{7}}}\:=\int\:\:\frac{{x}−\mathrm{2}}{\:\sqrt{\left({x}+\mathrm{2}\right)^{\mathrm{2}} −\mathrm{7}}}{dx}…
Question Number 48720 by Abdo msup. last updated on 27/Nov/18 $${calculate}\:\int_{\mathrm{0}} ^{\infty} \:\:\:\:\frac{{x}^{\mathrm{2}} \:−\mathrm{2}{cosx}+\mathrm{1}}{{x}^{\mathrm{4}} \:+{x}^{\mathrm{2}} \:+\mathrm{1}}{dx} \\ $$ Commented by Abdo msup. last updated on…
Question Number 48717 by Abdo msup. last updated on 27/Nov/18 $${let}\:\:{f}\left({x}\right)=\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{4}}} {ln}\left(\mathrm{1}+{xtant}\right){dt} \\ $$$$\left.\mathrm{1}\right)\:{find}\:{f}\left({x}\right)\:{at}\:{a}\:{simple}\:{form} \\ $$$$\left.\mathrm{2}\right){calculate}\:\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{4}}} {ln}\left(\mathrm{1}+\mathrm{2}{tan}\left({t}\right)\right){dt} \\ $$ Commented by maxmathsup by…
Question Number 48718 by Abdo msup. last updated on 27/Nov/18 $${let}\:{I}_{{n}} =\int_{\mathrm{0}} ^{\mathrm{1}} \left(\mathrm{1}−{t}^{\mathrm{2}} \right)^{{n}} {dt} \\ $$$$\left.\mathrm{1}\right)\:{calculate}\:{I}_{{n}} \:{by}\:{recurrence} \\ $$$$\left.\mathrm{2}\right){find}\:{the}\:{value}\:{of}\:\sum_{{k}=\mathrm{0}} ^{{n}} \:\frac{\left(−\mathrm{1}\right)^{{k}} }{\mathrm{2}{k}+\mathrm{1}}{C}_{{n}} ^{{k}}…
Question Number 48715 by Abdo msup. last updated on 27/Nov/18 $$\left.\mathrm{1}\right)\:{find}\:\:{f}\left(\lambda\right)\:=\int_{\mathrm{0}} ^{\mathrm{1}} \:\:\frac{{dx}}{\mathrm{2}+{e}^{−\lambda{x}} }\:\:{with}\:\lambda>\mathrm{0} \\ $$$$\left.\mathrm{2}\right){calculate}\:\int_{\mathrm{0}} ^{\mathrm{1}} \:\:\:\:\frac{{x}}{\left(\mathrm{2}+{e}^{−\lambda{x}} \right)^{\mathrm{2}} }{dx} \\ $$$$\left.\mathrm{3}\right)\:{find}\:{the}\:{value}\:{of}\:\int_{\mathrm{0}} ^{\mathrm{1}} \:\:\:\:\frac{{dx}}{\mathrm{2}\:+{e}^{−{x}\sqrt{\mathrm{3}}} }{dx}\:{and}\:\int_{\mathrm{0}}…
Question Number 48703 by cesar.marval.larez@gmail.com last updated on 27/Nov/18 Answered by tanmay.chaudhury50@gmail.com last updated on 27/Nov/18 $$\left.\mathrm{3}\right)\int{sin}^{\mathrm{3}} \left(\mathrm{2}{x}\right){cos}\left(\mathrm{2}{x}\right){dx} \\ $$$${t}={sin}\mathrm{2}{x}\:\:\:{dt}=\mathrm{2}{cos}\mathrm{2}{xdx} \\ $$$$=\frac{\mathrm{1}}{\mathrm{2}}\int{t}^{\mathrm{3}} {dt} \\ $$$$=\frac{\mathrm{1}}{\mathrm{2}}×\frac{{t}^{\mathrm{4}}…
Question Number 48667 by maxmathsup by imad last updated on 26/Nov/18 $${find}\:\int_{\mathrm{0}} ^{\mathrm{1}} \:\:\frac{{arctan}\left({x}\right)}{\:\sqrt{\mathrm{1}+{x}^{\mathrm{2}} }}\:{dx}\:. \\ $$ Commented by Abdo msup. last updated on 02/Dec/18…
Question Number 48643 by cesar.marval.larez@gmail.com last updated on 26/Nov/18 Commented by maxmathsup by imad last updated on 26/Nov/18 $$\left.\mathrm{1}\right){let}\:\:{I}\:=\int\:\:\frac{\mathrm{3}{x}+\mathrm{2}}{\left({x}−\mathrm{1}\right)\left({x}−\mathrm{2}\right)\left({x}^{\mathrm{2}} \:+\mathrm{3}\right)}{dx}\:{let}\:{decompose}\:{F}\left({x}\right)=\frac{\mathrm{3}{x}+\mathrm{2}}{\left({x}−\mathrm{1}\right)\left({x}−\mathrm{2}\right)\left({x}^{\mathrm{2}} \:+\mathrm{3}\right)} \\ $$$${F}\left({x}\right)\:=\frac{{a}}{{x}−\mathrm{1}}\:+\frac{{b}}{{x}−\mathrm{2}}\:+\frac{{cx}+{d}}{{x}^{\mathrm{2}\:} +\mathrm{3}} \\…
Question Number 114161 by bemath last updated on 17/Sep/20 $$\:\int\:\frac{{dx}}{{x}^{\mathrm{4}} −\mathrm{5}{x}^{\mathrm{2}} −\mathrm{16}} \\ $$ Answered by Olaf last updated on 17/Sep/20 $$−\frac{\mathrm{2}}{\:\sqrt{\mathrm{89}}}\frac{\mathrm{arctan}\left(\frac{\sqrt{\mathrm{2}}{x}}{\:\sqrt{\sqrt{\mathrm{89}}−\mathrm{5}}}\right)}{\:\sqrt{\sqrt{\mathrm{89}}−\mathrm{5}}}−\frac{\mathrm{2}}{\:\sqrt{\mathrm{89}}}\frac{\mathrm{arctan}\left(\frac{\sqrt{\mathrm{2}}{x}}{\:\sqrt{\sqrt{\mathrm{89}}+\mathrm{5}}}\right)}{\:\sqrt{\sqrt{\mathrm{89}}+\mathrm{5}}} \\ $$ Answered…
Question Number 114146 by Eric002 last updated on 17/Sep/20 $${prove} \\ $$$$\int_{\mathrm{0}} ^{\mathrm{1}} \frac{{t}^{{n}+\mathrm{2}} \phi\left({t},\mathrm{1},{n}+\mathrm{2}\right)+{ln}\left(\mathrm{1}−{t}\right)+{t}\:{H}_{{n}+\mathrm{1}} }{{t}\left({t}−\mathrm{1}\right)}{dt} \\ $$$$=\frac{{H}_{{n}+\mathrm{1}} ^{\left(\mathrm{2}\right)} −\left({H}_{{n}} \right)^{\mathrm{2}} }{\mathrm{2}} \\ $$ Commented…