Menu Close

x-5-2-1-x-3-2-dx-




Question Number 27615 by NECx last updated on 10/Jan/18
∫x^(5/2) (1−x)^(3/2) dx
$$\int{x}^{\mathrm{5}/\mathrm{2}} \left(\mathrm{1}−{x}\right)^{\mathrm{3}/\mathrm{2}} {dx} \\ $$
Commented by abdo imad last updated on 11/Jan/18
 let do the changement   x =sin^2 t  ∫x^(5/2)  (1−x)^(3/2) dx=  2∫ sin^5 t cos^3 t sint cost dt  = 2 ∫ sin^6 t cos^4 t dt  =2 ∫ (sint cost)^4  sin^2 t dt= (2/2^4 ) ∫ (sin(2t))^4 sin^2 tdt  = (1/8) ∫ (sin(2t))^4  sin^2 tdt  after we do the linesrisaton  sin2t)^4 = (((e^(i2t ) −e^(−i2t) )/(2i)) )^4    and   sin^2 t =(((e^(it)  −e^(−it) )/(2i)))^2 .....
$$\:{let}\:{do}\:{the}\:{changement}\:\:\:{x}\:={sin}^{\mathrm{2}} {t} \\ $$$$\int{x}^{\frac{\mathrm{5}}{\mathrm{2}}} \:\left(\mathrm{1}−{x}\right)^{\frac{\mathrm{3}}{\mathrm{2}}} {dx}=\:\:\mathrm{2}\int\:{sin}^{\mathrm{5}} {t}\:{cos}^{\mathrm{3}} {t}\:{sint}\:{cost}\:{dt} \\ $$$$=\:\mathrm{2}\:\int\:{sin}^{\mathrm{6}} {t}\:{cos}^{\mathrm{4}} {t}\:{dt} \\ $$$$=\mathrm{2}\:\int\:\left({sint}\:{cost}\right)^{\mathrm{4}} \:{sin}^{\mathrm{2}} {t}\:{dt}=\:\frac{\mathrm{2}}{\mathrm{2}^{\mathrm{4}} }\:\int\:\left({sin}\left(\mathrm{2}{t}\right)\right)^{\mathrm{4}} {sin}^{\mathrm{2}} {tdt} \\ $$$$=\:\frac{\mathrm{1}}{\mathrm{8}}\:\int\:\left({sin}\left(\mathrm{2}{t}\right)\right)^{\mathrm{4}} \:{sin}^{\mathrm{2}} {tdt}\:\:{after}\:{we}\:{do}\:{the}\:{linesrisaton} \\ $$$$\left.{sin}\mathrm{2}{t}\right)^{\mathrm{4}} =\:\left(\frac{{e}^{{i}\mathrm{2}{t}\:} −{e}^{−{i}\mathrm{2}{t}} }{\mathrm{2}{i}}\:\right)^{\mathrm{4}} \:\:\:{and}\:\:\:{sin}^{\mathrm{2}} {t}\:=\left(\frac{{e}^{{it}} \:−{e}^{−{it}} }{\mathrm{2}{i}}\right)^{\mathrm{2}} ….. \\ $$
Commented by NECx last updated on 11/Jan/18
please complete it ... Thanks
$${please}\:{complete}\:{it}\:…\:{Thanks} \\ $$

Leave a Reply

Your email address will not be published. Required fields are marked *