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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?

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### 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.

Replace the letter A in the integral ∫ A e^(2x^3 + x^2) dx so that the integral evaluates to (1/2)e^(2x^3 + x^2) + C.

Solve for the constant A in the integral ∫ A e^(2x^3 + x^2) dx to make the result equal to (1/2)e^(2x^3 + x^2) + C. This solution includes substitution techniques for exponential integration, ideal for students in Grades 11-12.

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