The integral given in the image is:

$\int x \sqrt{x + 6} \, dx$

To solve this, we will perform a substitution. Let’s define:

$u = x + 6$

Now, differentiate both sides with respect to $x$:

$du = dx$ $x = u - 6$

Substitute these into the integral:

$\int (u - 6) \sqrt{u} \, du$

Now, break this into two separate integrals:

$\int u^{1/2}(u - 6) \, du = \int u^{3/2} \, du - 6 \int u^{1/2} \, du$

These integrals can now be solved using the power rule. The power rule for integration is:

\int u^n \, du = \frac{u^{n+1}}{n+1} + C \quad \text{(where n \neq -1)}

Apply the power rule to each term:

$\int u^{3/2} \, du = \frac{u^{5/2}}{5/2} = \frac{2}{5} u^{5/2}$ $6 \int u^{1/2} \, du = 6 \cdot \frac{u^{3/2}}{3/2} = 4 u^{3/2}$

Thus, the integral becomes:

$\frac{2}{5} u^{5/2} - 4 u^{3/2} + C$

Finally, substitute $u = x + 6$ back into the solution:

$\frac{2}{5} (x + 6)^{5/2} - 4 (x + 6)^{3/2} + C$

This is the final solution to the integral.

Would you like more details or have any questions about this solution?

Related Questions:

1. How can we verify if this integral is correct?
2. What is the power rule of integration and when is it used?
3. Can the substitution method be applied to all integrals?
4. How would the answer change if the integral had a different constant instead of 6?
5. How can you apply integration by parts to solve similar problems?

Tip: Substitution is one of the most powerful methods for solving integrals, especially when simplifying expressions involving square roots.