Q1Advanced Engineering Mathematics

The Laplace Transform is fundamentally defined as the infinite integral $L\{f(t)\}={\int }_0^{\infty }{e}^{-st}f(t)dt$. For the given piecewise function, which is non-zero only strictly between $0$ and $2\pi$, the infinite upper bound of integration becomes finite at $2\pi$. We substitute $f(t)=\cos (t)$ and use the standard exponential-trigonometric integral formula to obtain $\frac{s(1-e^{-2\pi s})}{s^2+1}$.