The root mean square value of a signal (x(t), x_{mathrm{rms}}), is defined as [x_{mathrm{rms}}=left{lim _{T ightarrow

The root mean square value of a signal \(x(t), x_{\mathrm{rms}}\), is defined as

\[x_{\mathrm{rms}}=\left\{\lim _{T \rightarrow \infty} \frac{1}{T} \int_{0}^{T} x^{2}(t) d t\right\}^{1 / 2}\]

Using this definition, find the root mean square values of the displacement \(\left(x_{\mathrm{rms}}\right)\), velocity \(\left(\dot{x}_{\text {rms }}\right.\) ), and acceleration ( \(\left.\ddot{x}_{\text {rms }}\right)\) corresponding to \(x(t)=X \cos \omega t\).