To solve the integral $\int (2x + 1)e^{x^2 + x} \, dx$, we notice that the integrand suggests a substitution based on the exponent of the exponential term.

Step 1: Substitution

Let: $u = x^2 + x$ Then: $\frac{du}{dx} = 2x + 1 \quad \implies \quad du = (2x + 1) \, dx$

This substitution simplifies the integral: $\int (2x + 1)e^{x^2 + x} \, dx = \int e^u \, du$

Step 2: Integrate with respect to $u$

The integral of $e^u$ with respect to $u$ is straightforward: $\int e^u \, du = e^u + C$

Step 3: Back-substitute $u = x^2 + x$

Returning to the original variable: $e^u + C = e^{x^2 + x} + C$

Final Answer:

$\int (2x + 1)e^{x^2 + x} \, dx = e^{x^2 + x} + C$

---

Would you like further details or explanations on any step? 😊

Here are 5 related questions for further exploration:

1. How would the solution change if the exponent were $x^2 - x$ instead?
2. Can substitution always be used in integrals of exponential functions? Why or why not?
3. What happens if the integrand included an additional factor, such as $x^3$?
4. How do you handle integrals where the substitution doesn’t directly simplify the differential?
5. Could you compute $\int e^{x^2 + x} \, dx$ without the factor $(2x+1)$? Why is it more challenging?

Tip: Always check for patterns in the derivative of one part of the function to simplify substitution problems!