Question Number 55777 by gunawan last updated on 04/Mar/19
$$\underset{\:\mathrm{0}} {\overset{\pi} {\int}}\:\:\mathrm{cos}^{\mathrm{7}} {x}\:\mathrm{sin}\:\mathrm{3}{x}\:{dx}\:=\:\mathrm{0} \\ $$
Answered by tanmay.chaudhury50@gmail.com last updated on 04/Mar/19
$${I}=\int_{\mathrm{0}} ^{\pi} \left\{{cos}\left(\pi−{x}\right)\right\}^{\mathrm{7}} ×{sin}\mathrm{3}\left(\pi−{x}\right){dx} \\ $$$$=\int_{\mathrm{0}} ^{\pi} \left(−{cos}^{\mathrm{7}} {x}\right)×{sin}\left(\mathrm{3}\pi−\mathrm{3}{x}\right){dx} \\ $$$$=−\int_{\mathrm{0}} ^{\pi} \left({cos}^{\mathrm{7}} {x}\right)×{sin}\mathrm{3}{x} \\ $$$$=−{I} \\ $$$$\mathrm{2}{I}=\mathrm{0} \\ $$$${I}=\mathrm{0} \\ $$
Commented by gunawan last updated on 04/Mar/19
$$\mathrm{Great}\:\mathrm{Sir} \\ $$$$\mathrm{Thank}\:\mathrm{you} \\ $$