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Category: Integration

let-f-a-0-1-sin-2x-1-ax-2-dx-with-a-lt-1-1-approximate-f-a-by-a-polynom-2-find-the-value-perhaps-not-exact-of-0-1-sin-2x-1-2x-2-dx-3-let-g-a-0-1-x-2-s

Question Number 61328 by maxmathsup by imad last updated on 01/Jun/19 $${let}\:{f}\left({a}\right)\:=\int_{\mathrm{0}} ^{\mathrm{1}} \:\:\:\frac{{sin}\left(\mathrm{2}{x}\right)}{\mathrm{1}+{ax}^{\mathrm{2}} }\:{dx}\:\:{with}\:\:\mid{a}\mid<\mathrm{1} \\ $$$$\left.\mathrm{1}\right)\:{approximate}\:{f}\left({a}\right)\:{by}\:{a}\:{polynom} \\ $$$$\left.\mathrm{2}\right)\:{find}\:{the}\:{value}\:\:\left({perhaps}\:{not}\:{exact}\right)\:{of}\:\int_{\mathrm{0}} ^{\mathrm{1}} \:\:\frac{{sin}\left(\mathrm{2}{x}\right)}{\mathrm{1}+\mathrm{2}{x}^{\mathrm{2}} }\:{dx} \\ $$$$\left.\mathrm{3}\right)\:{let}\:{g}\left({a}\right)\:=\:\int_{\mathrm{0}} ^{\mathrm{1}}…

Question-192388

Question Number 192388 by sudipmoi last updated on 16/May/23 Answered by aleks041103 last updated on 22/May/23 $${p}>\mathrm{0} \\ $$$$\int_{\mathrm{1}} ^{\:\infty} \frac{{sin}\left({x}\right){dx}}{{x}^{{p}} }=\left[−\frac{{cos}\left({x}\right)}{{x}^{{p}} }\right]_{\mathrm{1}} ^{\infty} −\int_{\mathrm{1}}…

B-7-3-2-3-B-betha-function-

Question Number 126803 by john_santu last updated on 24/Dec/20 $$\:\:{B}\left(\frac{\mathrm{7}}{\mathrm{3}},\frac{\mathrm{2}}{\mathrm{3}}\right)\:=? \\ $$$${B}\:=\:{betha}\:{function}\: \\ $$ Answered by Dwaipayan Shikari last updated on 24/Dec/20 $${B}\left(\frac{\mathrm{7}}{\mathrm{3}},\frac{\mathrm{2}}{\mathrm{3}}\right)=\frac{\Gamma\left(\frac{\mathrm{7}}{\mathrm{3}}\right)\Gamma\left(\frac{\mathrm{2}}{\mathrm{3}}\right)}{\Gamma\left(\mathrm{3}\right)}=\frac{\Gamma\left(\frac{\mathrm{1}}{\mathrm{3}}\right)\Gamma\left(\frac{\mathrm{2}}{\mathrm{3}}\right).\frac{\mathrm{4}}{\mathrm{3}}.\frac{\mathrm{1}}{\mathrm{3}}}{\mathrm{2}!}=\frac{\mathrm{2}}{\mathrm{9}}.\frac{\pi}{{sin}\frac{\pi}{\mathrm{3}}}=\frac{\mathrm{4}\pi}{\mathrm{9}\sqrt{\mathrm{3}}} \\ $$…

0-x-e-x-4-dx-

Question Number 126788 by john_santu last updated on 24/Dec/20 $$\:\sigma\:=\:\underset{\mathrm{0}} {\overset{\:\:\:\:\:\infty} {\int}}\sqrt{{x}}\:{e}^{−{x}/\mathrm{4}} \:{dx}\:=\:?\: \\ $$ Answered by Ar Brandon last updated on 24/Dec/20 $$\mathrm{x}=\mathrm{u}^{\mathrm{2}} \:\Rightarrow\:\mathrm{dx}=\mathrm{2udu}…

x-2-4-x-2-4-2-dx-

Question Number 61240 by Tawa1 last updated on 30/May/19 $$\int\:\frac{\mathrm{x}^{\mathrm{2}\:} −\:\mathrm{4}}{\left(\mathrm{x}^{\mathrm{2}} \:+\:\mathrm{4}\right)^{\mathrm{2}} }\:\mathrm{dx} \\ $$ Commented by maxmathsup by imad last updated on 31/May/19 $${let}\:{A}\:=\int\:\:\frac{{x}^{\mathrm{2}}…