1. **Problem statement:** Find the Laplace transform of the function $f(t) = t \sqrt{\sin 2t}$. 2. **Recall the Laplace transform definition:** The Laplace transform of a function $f(t)$ is given by $$\mathcal{L}\{f(t)\} = \int_0^{\infty} e^{-st} f(t) \, dt$$ where $s$ is a complex number parameter. 3. **Analyze the function:** Here, $f(t) = t \sqrt{\sin 2t} = t (\sin 2t)^{1/2}$. This is not a standard Laplace transform form and involves a non-integer power of a trigonometric function, which complicates direct integration. 4. **Consider properties and simplifications:** - The Laplace transform of $t$ is $\frac{1}{s^2}$. - The Laplace transform of $\sin 2t$ is $\frac{2}{s^2 + 4}$. - However, the square root of $\sin 2t$ does not have a simple closed form Laplace transform. 5. **Conclusion:** The Laplace transform of $t \sqrt{\sin 2t}$ does not have a standard closed form and would require advanced special functions or numerical methods to evaluate. **Final answer:** The Laplace transform of $f(t) = t \sqrt{\sin 2t}$ cannot be expressed in a simple closed form using elementary functions.