$$int_-fracpi6^fracpi6left(int_0^6cos(3 heta) r dr ight) d heta$$

And I obtained an answer of $frac112pi$. At the finish of the difficulty, I got$$ frac14left(frac16pi + 6sinleft(fracpi6^2 ight) - frac14left(frac-pi6+6sin(-pi) ight) ight) $$

which need to be $frac112pi$ which I am unsure if it is best or not.




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$$ int_-fracpi6^fracpi6left(int_0^6cos(3 heta) r katifund.orgrm dr ight) katifund.orgrm d heta$$ $$=18int_-fracpi6^fracpi6cos^2(3 heta) katifund.orgrm d heta$$Because the integrand also is an also feature, we have$$36int_0^fracpi6cos^2(3 heta) katifund.orgrm d heta$$Now we deserve to usage the fact that$$int cos^2(ax) katifund.orgrm dx=fracx2+frac14asin(2ax)+C$$Wbelow in our case $a=3$, therefore$$36left(fracpi12+frac112sinleft(pi ight) ight)=frac36pi12=3pi$$


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