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To solve the integral \(\int x \sqrt{x + 6} \, dx\), we can use substitution to simplify it.

1. **Substitute** \( u = x + 6 \), so that \( du = dx \) and \( x = u - 6 \).
   
   Now rewrite the integral in terms of \( u \):
   \[
   \int x \sqrt{x + 6} \, dx = \int (u - 6) \sqrt{u} \, du
   \]

2. **Expand the Integral** by distributing \( \sqrt{u} \):
   \[
   = \int (u - 6) u^{1/2} \, du = \int \left( u^{3/2} - 6u^{1/2} \right) du
   \]

3. **Integrate Each Term**:
   - For \( \int u^{3/2} \, du \):
     \[
     \int u^{3/2} \, du = \frac{2}{5} u^{5/2}
     \]
   - For \( \int 6u^{1/2} \, du \):
     \[
     \int 6u^{1/2} \, du = 6 \cdot \frac{2}{3} u^{3/2} = 4 u^{3/2}
     \]

   Putting these results together:
   \[
   \int \left( u^{3/2} - 6u^{1/2} \right) du = \frac{2}{5} u^{5/2} - 4 u^{3/2} + C
   \]

4. **Substitute back** \( u = x + 6 \):
   \[
   = \frac{2}{5} (x + 6)^{5/2} - 4 (x + 6)^{3/2} + C
   \]

5. **Compare with the Options**:
   None of the options match exactly in terms of this simplified form. However, we can simplify further if needed to see if any options might represent equivalent expressions. 

The correct answer appears closest to one of the options if the simplified form matches an equivalent representation.

Integrate x√(x + 6) dx and select the correct answer from the given options.

Find the solution to the integral ∫x√(x + 6) dx by using substitution. This calculus problem involves algebraic manipulation and integration techniques. Suitable for college-level calculus students.

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