Answer to Question #223827 in Chemical Engineering for Lokika

Question

∫0^{π}cos3t 𝛿(t-2π) dt

Answer 1

$\int _0^{\pi }\cos \left(3t\right)\left(t-2\pi \right)dt\\ =\int _0^{\pi }t\cos \left(3t\right)-2\pi \cos \left(3t\right)dt\\ =t\cos \left(3t\right)-2\pi \cos \left(3t\right)\\ \mathrm{Apply\:the\:Sum\:Rule}:\quad \int f\left(x\right)\pm g\left(x\right)dx=\int f\left(x\right)dx\pm \int g\left(x\right)dx\\ =\int _0^{\pi }t\cos \left(3t\right)dt-\int _0^{\pi }2\pi \cos \left(3t\right)dt\\ =-\frac{2}{9}-0\\ =-\frac{2}{9}$