Question #223827

∫0^{π}cos3t 𝛿(t-2π) dt


1
Expert's answer
2021-08-16T02:20:32-0400

0πcos(3t)(t2π)dt=0πtcos(3t)2πcos(3t)dt=tcos(3t)2πcos(3t)ApplytheSumRule:f(x)±g(x)dx=f(x)dx±g(x)dx=0πtcos(3t)dt0π2πcos(3t)dt=290=29\int _0^{\pi }\cos \left(3t\right)\left(t-2\pi \right)dt\\ =\int _0^{\pi }t\cos \left(3t\right)-2\pi \cos \left(3t\right)dt\\ =t\cos \left(3t\right)-2\pi \cos \left(3t\right)\\ \mathrm{Apply\:the\:Sum\:Rule}:\quad \int f\left(x\right)\pm g\left(x\right)dx=\int f\left(x\right)dx\pm \int g\left(x\right)dx\\ =\int _0^{\pi }t\cos \left(3t\right)dt-\int _0^{\pi }2\pi \cos \left(3t\right)dt\\ =-\frac{2}{9}-0\\ =-\frac{2}{9}


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