∫0^{π}cos3t 𝛿(t-2π) dt
∫0πcos(3t)(t−2π)dt=∫0πtcos(3t)−2πcos(3t)dt=tcos(3t)−2πcos(3t)Apply the Sum Rule:∫f(x)±g(x)dx=∫f(x)dx±∫g(x)dx=∫0πtcos(3t)dt−∫0π2πcos(3t)dt=−29−0=−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}∫0πcos(3t)(t−2π)dt=∫0πtcos(3t)−2πcos(3t)dt=tcos(3t)−2πcos(3t)ApplytheSumRule:∫f(x)±g(x)dx=∫f(x)dx±∫g(x)dx=∫0πtcos(3t)dt−∫0π2πcos(3t)dt=−92−0=−92
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