I am trying to find the expression for energy dissipation per cycle of a damper (force = cx') using
$$\frac{dW}{dt} = fx' = c(x')^2$$
My professor has used the substitution $x = X \sin(wt)$ to derive an expression for $x'$ in terms of $t$ to get the final expression:
$$W = \pi cw(X^2)$$
I tried the same integral, instead using the complex phasor substitution $x = Xe^{iwt}$, which gave me: $$W = c(j^2)(w^2)(X^2) \int_0^{2 \pi / w}e^{2jwt} dt$$
This integral evaluates to zero and I can't see my error, if someone could enlighten me that would be great!