Evaluate the integral e^-x cos3x dx between x=0 and x=infinity Solution Since si

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Since sine and cosine are bounded between -1 and 1 inclusive, one obtains the crude bound -3 - 1 = 3 sin(3x) - cos(3x) = 3 + 1 ==> -4 = 3 sin(3x) - cos(3x) = 4 for all x ==> (-4/10) e^(-x) = (1/10) e^(-x) [3 sin(3x) - cos(3x)] = (4/10) e^(-x) for all x Since lim(x?8) (±4/10) e^(-x) = 0, we conclude that lim(x?8) (1/10) e^(-x) [3 sin(3x) - cos(3x)] = 0. In summary: ?(x = 0 to 8) e^(-x) cos(3x) dx = lim(t?8) ?(x = 0 to t) e^(-x) cos(3x) dx = lim(t?8) (1/10) e^(-x) [3 sin(3x) - cos(3x)] {for x = 0 to t} = lim(t?8) {(1/10) e^(-t) [3 sin(3t) - cos(3t)] - (1/10) (0 - 1)} = 0 + 1/10, by the above work = 1/10.

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