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

∫_0 ^π e^(sin^2 x) Cos^3 xdx

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

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

Commented by Hitarth Rana last updated on 24/Sep/18

Perfect...★★★★★

$${Perfect}...\bigstar\bigstar\bigstar\bigstar\bigstar \\ $$

Commented by LYCON TRIX last updated on 24/Sep/18

Thanks a lot gentlemen

$$\mathrm{Thanks}\:\mathrm{a}\:\mathrm{lot}\:\mathrm{gentlemen} \\ $$

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

thank you...

$${thank}\:{you}... \\ $$

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