Question Number 44305 by abdo.msup.com last updated on 26/Sep/18 $${let}\:{f}\left({a}\right)\:=\int_{\mathrm{0}} ^{\infty} \:{ln}\left(\mathrm{1}+\frac{{a}^{\mathrm{2}} }{{x}^{\mathrm{2}} }\right){dx} \\ $$$$\left.\mathrm{1}\right)\:{find}\:{a}\:{explicit}\:{form}\:{of}\:{f}\left({x}\right) \\ $$$$\left.\mathrm{2}\right){find}\:\int_{\mathrm{0}} ^{\infty} \:{ln}\left(\mathrm{1}+\frac{\mathrm{1}}{{x}^{\mathrm{2}} }\right){dx} \\ $$$$\left.\mathrm{3}\right){calculate}\:\int_{\mathrm{0}} ^{\infty} \:{ln}\left(\mathrm{1}+\frac{\mathrm{2}}{{x}^{\mathrm{2}}…
Question Number 44306 by abdo.msup.com last updated on 26/Sep/18 $${find}\:\int\:\:\frac{{dt}}{\left({t}+\mathrm{1}\right)\sqrt{{t}}\:+{t}\sqrt{{t}+\mathrm{1}}} \\ $$$$\left.\mathrm{2}\right)\:{calculate}\:\:\int_{\mathrm{1}} ^{\mathrm{3}} \:\:\frac{{dt}}{\left({t}+\mathrm{1}\right)\sqrt{{t}}+{t}\sqrt{{t}+\mathrm{1}}} \\ $$ Commented by maxmathsup by imad last updated on 29/Sep/18…
Question Number 44304 by abdo.msup.com last updated on 26/Sep/18 $${find}\:{f}\left({a}\right)=\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{4}}} \:\:\frac{{dx}}{\mathrm{1}+{acos}^{\mathrm{2}} {x}} \\ $$$${a}\:{from}\:{R}. \\ $$ Commented by maxmathsup by imad last updated on…
Question Number 109839 by john santu last updated on 25/Aug/20 $$\:\frac{{JS}}{\approx\heartsuit\approx} \\ $$$$\left(\mathrm{1}\right)\:\int\:\frac{\sqrt{{x}+\mathrm{1}}−\sqrt{{x}−\mathrm{1}}}{\:\sqrt{{x}+\mathrm{1}}+\sqrt{{x}−\mathrm{1}}}\:{dx} \\ $$$$\left(\mathrm{2}\right)\:\int\:\frac{\sqrt{\mathrm{tan}\:{x}}}{\mathrm{1}+\sqrt{\mathrm{tan}\:{x}}\:}\:{dx}\: \\ $$ Commented by Her_Majesty last updated on 25/Aug/20 $$\left(\mathrm{1}\right)\:=\int{x}−\sqrt{{x}^{\mathrm{2}}…
Question Number 109838 by mnjuly1970 last updated on 25/Aug/20 $$ \\ $$$$ \\ $$$$\:\:\:\:{please}\:{prove}::: \\ $$$$\int_{\mathrm{0}} ^{\:\mathrm{1}} \frac{\mathrm{1}}{\:\sqrt{\mathrm{1}−{x}}}{log}\left(\frac{{x}}{\mathrm{1}−{x}}\right){dx}\:=\mathrm{4}{log}\left(\mathrm{2}\right)\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\: \\ $$$$ \\ $$$$ \\ $$ Answered…
Question Number 44302 by abdo.msup.com last updated on 26/Sep/18 $${calculate}\:\int_{\mathrm{0}} ^{\infty} \:\:\frac{{dx}}{\left({x}+\mathrm{1}\right)\left({x}+\mathrm{2}\right)\left({x}+\mathrm{3}\right)} \\ $$ Commented by maxmathsup by imad last updated on 28/Sep/18 $${let}\:{I}\:=\int_{\mathrm{0}} ^{\infty}…
Question Number 175351 by cortano1 last updated on 28/Aug/22 $$\:\:\underset{\mathrm{0}} {\overset{\pi/\mathrm{2}} {\int}}\:\frac{\mathrm{sin}\:^{\mathrm{3}} {x}}{\mathrm{sin}\:{x}+\mathrm{cos}\:{x}}\:{dx}\:=? \\ $$ Answered by som(math1967) last updated on 28/Aug/22 $${I}=\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \frac{{sin}^{\mathrm{3}}…
Question Number 175343 by rexford last updated on 27/Aug/22 $$\int_{\mathrm{0}} ^{\infty} \frac{{x}^{\mathrm{5}} }{\:\sqrt{\mathrm{3}−{x}}}{dx} \\ $$ Commented by BaliramKumar last updated on 28/Aug/22 $${I}\:{think}\:{Q}.\:{will}\:{be}\:\:\:\underset{\mathrm{0}} {\overset{\mathrm{3}} {\int}}\frac{\:{x}^{\mathrm{5}}…
Question Number 44264 by pramid last updated on 25/Sep/18 $$\int\frac{\mathrm{1}}{\left(\mathrm{x}^{\mathrm{2}} +\mathrm{2x}+\mathrm{5}\right)^{\mathrm{2}} }\mathrm{dx} \\ $$ Commented by maxmathsup by imad last updated on 25/Sep/18 $${let}\:{I}\:=\:\int\:\frac{{dx}}{\left({x}^{\mathrm{2}} \:+\mathrm{2}{x}\:+\mathrm{5}\right)^{\mathrm{2}}…
Question Number 109801 by mathmax by abdo last updated on 25/Aug/20 $$\mathrm{calculste}\:\int_{\mathrm{0}} ^{\mathrm{1}} \sqrt{\mathrm{1}+\mathrm{x}^{\mathrm{6}} }\mathrm{dx} \\ $$ Commented by Dwaipayan Shikari last updated on 25/Aug/20…