1. The problem is to find the integral $$\int x^3 \cos(x^4 + 2) \, dx$$. 2. Notice that the argument of the cosine function is $$x^4 + 2$$, and its derivative is $$4x^3$$, which is similar to the factor $$x^3$$ outside. 3. Use substitution: let $$u = x^4 + 2$$. 4. Then, $$\frac{du}{dx} = 4x^3$$, so $$du = 4x^3 dx$$. 5. Solve for $$x^3 dx$$: $$x^3 dx = \frac{du}{4}$$. 6. Substitute into the integral: $$\int x^3 \cos(x^4 + 2) \, dx = \int \cos(u) \frac{du}{4} = \frac{1}{4} \int \cos(u) \, du$$. 7. Integrate $$\cos(u)$$: $$\int \cos(u) \, du = \sin(u) + C$$. 8. Substitute back $$u = x^4 + 2$$: $$\frac{1}{4} \sin(x^4 + 2) + C$$. Final answer: $$\int x^3 \cos(x^4 + 2) \, dx = \frac{1}{4} \sin(x^4 + 2) + C$$