1. The problem appears to involve differentiating or integrating expressions involving $\sin x$ and $t$. However, the input is unclear and incomplete. 2. Assuming you want to differentiate $L = \sin x$ with respect to $x$, the derivative is: $$\frac{dL}{dx} = \cos x$$ 3. If you want to integrate $\frac{1}{3t^2}$ with respect to $t$, then: $$\int \frac{1}{3t^2} dt = \frac{1}{3} \int t^{-2} dt = \frac{1}{3} \left(-t^{-1}\right) + C = -\frac{1}{3t} + C$$ 4. Please clarify if you want differentiation or integration or something else for these expressions.