1. **State the problem:** Simplify or analyze the expression $x^3 \ln \sqrt{x^2+1}$. 2. **Recall relevant formulas and rules:** - The logarithm of a square root can be rewritten using the property $\ln \sqrt{a} = \frac{1}{2} \ln a$. - The expression involves a product of $x^3$ and a logarithm function. 3. **Rewrite the logarithm:** $$\ln \sqrt{x^2+1} = \frac{1}{2} \ln (x^2+1)$$ 4. **Substitute back into the expression:** $$x^3 \ln \sqrt{x^2+1} = x^3 \cdot \frac{1}{2} \ln (x^2+1) = \frac{x^3}{2} \ln (x^2+1)$$ 5. **Interpretation:** The expression is simplified to $$\frac{x^3}{2} \ln (x^2+1)$$ which is easier to work with for further calculus operations like differentiation or integration if needed. **Final answer:** $$\boxed{\frac{x^3}{2} \ln (x^2+1)}$$