Question Number 27615 by NECx last updated on 10/Jan/18
$$\int{x}^{\mathrm{5}/\mathrm{2}} \left(\mathrm{1}−{x}\right)^{\mathrm{3}/\mathrm{2}} {dx} \\ $$
Commented by abdo imad last updated on 11/Jan/18
$$\:{let}\:{do}\:{the}\:{changement}\:\:\:{x}\:={sin}^{\mathrm{2}} {t} \\ $$$$\int{x}^{\frac{\mathrm{5}}{\mathrm{2}}} \:\left(\mathrm{1}−{x}\right)^{\frac{\mathrm{3}}{\mathrm{2}}} {dx}=\:\:\mathrm{2}\int\:{sin}^{\mathrm{5}} {t}\:{cos}^{\mathrm{3}} {t}\:{sint}\:{cost}\:{dt} \\ $$$$=\:\mathrm{2}\:\int\:{sin}^{\mathrm{6}} {t}\:{cos}^{\mathrm{4}} {t}\:{dt} \\ $$$$=\mathrm{2}\:\int\:\left({sint}\:{cost}\right)^{\mathrm{4}} \:{sin}^{\mathrm{2}} {t}\:{dt}=\:\frac{\mathrm{2}}{\mathrm{2}^{\mathrm{4}} }\:\int\:\left({sin}\left(\mathrm{2}{t}\right)\right)^{\mathrm{4}} {sin}^{\mathrm{2}} {tdt} \\ $$$$=\:\frac{\mathrm{1}}{\mathrm{8}}\:\int\:\left({sin}\left(\mathrm{2}{t}\right)\right)^{\mathrm{4}} \:{sin}^{\mathrm{2}} {tdt}\:\:{after}\:{we}\:{do}\:{the}\:{linesrisaton} \\ $$$$\left.{sin}\mathrm{2}{t}\right)^{\mathrm{4}} =\:\left(\frac{{e}^{{i}\mathrm{2}{t}\:} −{e}^{−{i}\mathrm{2}{t}} }{\mathrm{2}{i}}\:\right)^{\mathrm{4}} \:\:\:{and}\:\:\:{sin}^{\mathrm{2}} {t}\:=\left(\frac{{e}^{{it}} \:−{e}^{−{it}} }{\mathrm{2}{i}}\right)^{\mathrm{2}} ….. \\ $$
Commented by NECx last updated on 11/Jan/18
$${please}\:{complete}\:{it}\:…\:{Thanks} \\ $$