0-pi-e-sin-2-x-Cos-3-xdx-

Question Number 44232 by LYCON TRIX last updated on 24/Sep/18

$$\int_{\mathrm{0}} ^{\pi} \mathrm{e}^{\mathrm{sin}^{\mathrm{2}} \mathrm{x}} \mathrm{Cos}^{\mathrm{3}} \mathrm{xdx} \\ $$

Answered by tanmay.chaudhury50@gmail.com last updated on 24/Sep/18

f(x)=e^(sin^2 x) cos^3 x f(Π−x)=e^({sin(Π−x)}^2 ) {cos(Π−x)}^3 =e^(sin^2 x) ×(−cosx)^3 =−f(x) so∫_0 ^Π e^(sin^2 x) cos^3 xdx=0

$${f}\left({x}\right)={e}^{{sin}^{\mathrm{2}} {x}} {cos}^{\mathrm{3}} {x} \\ $$$${f}\left(\Pi−{x}\right)={e}^{\left\{{sin}\left(\Pi−{x}\right)\right\}^{\mathrm{2}} } \left\{{cos}\left(\Pi−{x}\right)\right\}^{\mathrm{3}} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:={e}^{{sin}^{\mathrm{2}} {x}} ×\left(−{cosx}\right)^{\mathrm{3}} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:=−{f}\left({x}\right) \\ $$$${so}\int_{\mathrm{0}} ^{\Pi} {e}^{{sin}^{\mathrm{2}} {x}} {cos}^{\mathrm{3}} {xdx}=\mathrm{0} \\ $$

Commented by tanmay.chaudhury50@gmail.com last updated on 24/Sep/18