The Laplace transform of the function \(1 + t\) can be computed using the definition of the Laplace transform, which is given by \(\mathcal{L}[f(t)](s) := \int_0^\infty e^{-st}f(t)dt\). To find \(\mathcal{L}[1 + t]\), one must evaluate the integral \(\int_0^\infty e^{-st}(1 + t)dt\). This involves breaking the integral into two parts: \(\int_0^\infty e^{-st}dt\) and \(\int_0^\infty te^{-st}dt\), leading to the final result of \(\frac{1}{s^2}\) for the \(t\) term and \(\frac{1}{s}\) for the constant term, yielding \(\mathcal{L}[1 + t] = \frac{1}{s} + \frac{1}{s^2}\). This method does not rely on Laplace transform tables.
PREREQUISITESStudents of engineering, mathematics, and physics, as well as professionals working with differential equations and control systems, will benefit from this discussion on calculating the Laplace transform without reference tables.