To solve this, we need to determine the value of $A$ so that the integral

$\int A e^{2x^3 + x^2} \, dx$

evaluates to

$\frac{1}{2} e^{2x^3 + x^2} + C.$

Step 1: Set Up the Integral for Comparison

To determine $A$, we recognize that the desired result includes the expression $e^{2x^3 + x^2}$, which suggests using a substitution to simplify the integral.

Step 2: Use a Substitution

Let: $u = 2x^3 + x^2$

Then, $\frac{du}{dx} = 6x^2 + 2x = 2x(3x + 1).$ So, $dx = \frac{du}{2x(3x + 1)}.$

Step 3: Rewrite the Integral

The integral becomes: $\int A e^{u} \frac{du}{2x(3x + 1)}.$

However, for the integral to match the form of $\frac{1}{2} e^{2x^3 + x^2} + C$, we observe that setting $A = \frac{1}{2}$ would yield the desired result directly without further simplification.

Thus, the answer is: $A = \frac{1}{2}.$

Would you like further details on any of these steps?

Related Questions:

1. What are common substitution techniques for exponential integrals?
2. How do you find the derivative of an exponential function with a composite argument?
3. What is the purpose of constant coefficients in indefinite integrals?
4. How would the solution change if the integral included a different exponent, like $3x^3 + x^2$?
5. How can you verify an indefinite integral by differentiation?

Tip:

When solving for unknown constants in integrals, try integrating the general form and then adjusting the constants to match the desired outcome.