1. **State the problem:** We need to find the integral $$\int \sqrt{16x} \sin\left(1 + x^{32}\right) \, dx$$. 2. **Rewrite the integral:** Note that $$\sqrt{16x} = \sqrt{16} \sqrt{x} = 4 \sqrt{x}$$, so the integral becomes: $$\int 4 \sqrt{x} \sin\left(1 + x^{32}\right) \, dx = 4 \int x^{\frac{1}{2}} \sin\left(1 + x^{32}\right) \, dx$$. 3. **Consider substitution:** Let $$u = 1 + x^{32}$$. 4. **Find $$du$$:** $$du = 32 x^{31} dx \implies dx = \frac{du}{32 x^{31}}$$. 5. **Rewrite the integral in terms of $$u$$:** Substituting, we get: $$4 \int x^{\frac{1}{2}} \sin(u) \frac{du}{32 x^{31}} = \frac{4}{32} \int x^{\frac{1}{2} - 31} \sin(u) \, du = \frac{1}{8} \int x^{-\frac{61}{2}} \sin(u) \, du$$. 6. **Problem:** The integral still contains $$x$$ in terms of $$u$$, and $$x$$ cannot be easily expressed as a function of $$u$$ because $$u = 1 + x^{32}$$. 7. **Conclusion:** This integral does not have an elementary antiderivative expressible in terms of elementary functions due to the complicated composition of $$x^{32}$$ inside the sine and the factor $$\sqrt{16x}$$. 8. **Alternative:** The integral can be evaluated numerically or expressed in terms of special functions. **Final answer:** The integral $$\int \sqrt{16x} \sin\left(1 + x^{32}\right) \, dx$$ does not have a simple closed-form antiderivative in elementary functions.