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Question Number 55777 by gunawan last updated on 04/Mar/19

∫_( 0) ^π   cos^7 x sin 3x dx = 0

$$\underset{\:\mathrm{0}} {\overset{\pi} {\int}}\:\:\mathrm{cos}^{\mathrm{7}} {x}\:\mathrm{sin}\:\mathrm{3}{x}\:{dx}\:=\:\mathrm{0} \\ $$

Answered by tanmay.chaudhury50@gmail.com last updated on 04/Mar/19

I=∫_0 ^π {cos(π−x)}^7 ×sin3(π−x)dx  =∫_0 ^π (−cos^7 x)×sin(3π−3x)dx  =−∫_0 ^π (cos^7 x)×sin3x  =−I  2I=0  I=0

$${I}=\int_{\mathrm{0}} ^{\pi} \left\{{cos}\left(\pi−{x}\right)\right\}^{\mathrm{7}} ×{sin}\mathrm{3}\left(\pi−{x}\right){dx} \\ $$ $$=\int_{\mathrm{0}} ^{\pi} \left(−{cos}^{\mathrm{7}} {x}\right)×{sin}\left(\mathrm{3}\pi−\mathrm{3}{x}\right){dx} \\ $$ $$=−\int_{\mathrm{0}} ^{\pi} \left({cos}^{\mathrm{7}} {x}\right)×{sin}\mathrm{3}{x} \\ $$ $$=−{I} \\ $$ $$\mathrm{2}{I}=\mathrm{0} \\ $$ $${I}=\mathrm{0} \\ $$

Commented bygunawan last updated on 04/Mar/19

Great Sir  Thank you

$$\mathrm{Great}\:\mathrm{Sir} \\ $$ $$\mathrm{Thank}\:\mathrm{you} \\ $$

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