1. **State the problem:** We want to evaluate the integral $$\int x^2 \cos\left(x^3\right) \, dx.$$\n\n2. **Identify the method:** This integral suggests a substitution because the argument of cosine is $x^3$ and its derivative involves $x^2$.\n\n3. **Substitution:** Let $$u = x^3.$$ Then, $$du = 3x^2 \, dx \implies x^2 \, dx = \frac{du}{3}.$$\n\n4. **Rewrite the integral:** Substitute into the integral: $$\int x^2 \cos(x^3) \, dx = \int \cos(u) \cdot \frac{du}{3} = \frac{1}{3} \int \cos(u) \, du.$$\n\n5. **Integrate:** The integral of cosine is sine, so $$\frac{1}{3} \int \cos(u) \, du = \frac{1}{3} \sin(u) + C.$$\n\n6. **Back-substitute:** Replace $u$ with $x^3$ to get the final answer: $$\frac{1}{3} \sin(x^3) + C.$$