1. **Stating the problem:** Evaluate the integral $$\int (2x + 1) \cos(x^2 + x) \, dx$$. 2. **Formula and substitution rule:** When integrating a function of the form $$f(g(x)) \cdot g'(x)$$, we use substitution: let $$u = g(x)$$, then $$du = g'(x) dx$$, and the integral becomes $$\int f(u) \, du$$. 3. **Identify substitution:** Here, let $$u = x^2 + x$$. 4. **Compute $$du$$:** $$du = (2x + 1) dx$$. 5. **Rewrite the integral:** Substitute $$u$$ and $$du$$ into the integral: $$\int (2x + 1) \cos(x^2 + x) \, dx = \int \cos(u) \, du$$. 6. **Integrate:** The integral of $$\cos(u)$$ with respect to $$u$$ is $$\sin(u) + C$$. 7. **Back-substitute:** Replace $$u$$ with $$x^2 + x$$: $$\sin(x^2 + x) + C$$. **Final answer:** $$\int (2x + 1) \cos(x^2 + x) \, dx = \sin(x^2 + x) + C$$