1. **State the problem:** Find the inverse Laplace transform of the expression $$\sqrt{5-1} + \ln \sqrt{5^2 -1^2}$$. 2. **Simplify the expression inside the Laplace inverse:** Calculate each term: $$\sqrt{5-1} = \sqrt{4} = 2$$ $$5^2 - 1^2 = 25 - 1 = 24$$ $$\sqrt{24} = 2\sqrt{6}$$ So the expression becomes: $$2 + \ln(2\sqrt{6})$$ 3. **Simplify the logarithm:** Using log properties: $$\ln(2\sqrt{6}) = \ln 2 + \ln \sqrt{6} = \ln 2 + \frac{1}{2} \ln 6$$ 4. **Evaluate the constants:** Since all terms are constants, the entire expression is a constant: $$C = 2 + \ln 2 + \frac{1}{2} \ln 6$$ 5. **Recall the inverse Laplace transform of a constant:** $$\mathcal{L}^{-1}[C] = C \cdot \delta(t)$$ where $\delta(t)$ is the Dirac delta function. 6. **Final answer:** $$\mathcal{L}^{-1} \left[ \sqrt{5-1} + \ln \sqrt{5^2 -1^2} \right] = \left(2 + \ln 2 + \frac{1}{2} \ln 6 \right) \delta(t)$$