1. **State the problem:** Evaluate the integral $$\int \cos(10x) \cos(5x) \, dx$$. 2. **Use the product-to-sum formula:** For cosine functions, the product-to-sum identity is $$\cos A \cos B = \frac{1}{2} [\cos(A-B) + \cos(A+B)]$$. 3. **Apply the formula:** Let $A = 10x$ and $B = 5x$, then $$\cos(10x) \cos(5x) = \frac{1}{2} [\cos(10x - 5x) + \cos(10x + 5x)] = \frac{1}{2} [\cos(5x) + \cos(15x)]$$. 4. **Rewrite the integral:** $$\int \cos(10x) \cos(5x) \, dx = \int \frac{1}{2} [\cos(5x) + \cos(15x)] \, dx = \frac{1}{2} \int \cos(5x) \, dx + \frac{1}{2} \int \cos(15x) \, dx$$. 5. **Integrate each term:** - $$\int \cos(5x) \, dx = \frac{1}{5} \sin(5x) + C_1$$ - $$\int \cos(15x) \, dx = \frac{1}{15} \sin(15x) + C_2$$ 6. **Combine results:** $$\int \cos(10x) \cos(5x) \, dx = \frac{1}{2} \cdot \frac{1}{5} \sin(5x) + \frac{1}{2} \cdot \frac{1}{15} \sin(15x) + C$$ 7. **Simplify coefficients:** $$= \frac{1}{10} \sin(5x) + \frac{1}{30} \sin(15x) + C$$ **Final answer:** $$\int \cos(10x) \cos(5x) \, dx = \frac{1}{10} \sin(5x) + \frac{1}{30} \sin(15x) + C$$