To solve this problem, let's proceed with the following steps.

Problem Analysis

We need to evaluate the integral $\int A \left(3x^5 - 2\right)^5 \, dx$ and find the value of $A$ so that the result of this integral equals $\frac{1}{6} \left(3x^5 - 2\right)^6 + C.$

Solution Strategy

1. Set up the integral: We are given an expression inside the integral, and we are told that after integration, it should yield a certain form. To determine $A$, we can set up an equation based on the required result.

2. Use the Chain Rule in Integration: The integral has a term $(3x^5 - 2)^5$. This suggests that a $u$-substitution might work, where $u = 3x^5 - 2$.

3. Perform $u$-substitution:

   • Let $u = 3x^5 - 2$.
   • Then, $\frac{du}{dx} = 15x^4$, so $dx = \frac{du}{15x^4}$.

4. Rewrite the Integral: Substitute $u$ and $dx$ into the integral: $\int A \left(3x^5 - 2\right)^5 \, dx = \int A u^5 \cdot \frac{du}{15x^4}.$ Since $u = 3x^5 - 2$, the term $x^4$ will cancel out.