1. A body executes damped forced vibrations that are governed by the differential equation: (d^2x)/(dt^2) + 2k(dx)/(dt) + a^2x = e^-ktsinω t where k and a are constants. Solve the differential equation and analyze motion of the body for different values of ω. OR Evaluate (i) limx→ 0([(1+x)^1/x - e])/(x), (ii) ∫0^1 t^3/2(1-t)^3/2dt.
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