The integral provided in the image is:

$\int x^5 \left(x^6 + 9\right)^7 dx$

We can solve this integral using substitution. Let:

$u = x^6 + 9$

Then, differentiate both sides to find $du$:

$du = 6x^5 dx$

Now, we can rewrite the integral in terms of $u$:

$\int x^5 \left(x^6 + 9\right)^7 dx = \frac{1}{6} \int u^7 du$

The integral of $u^7$ is:

$\int u^7 du = \frac{u^8}{8}$

Therefore, the result of the integral becomes:

$\frac{1}{6} \times \frac{u^8}{8} = \frac{1}{48} \left(x^6 + 9\right)^8 + C$

Hence, the correct answer is:

(B) $\frac{1}{48} (x^6 + 9)^8 + C$

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

Here are some related questions:

1. How would you solve an integral with a different substitution method?
2. Can you verify this result using integration by parts?
3. What happens if we change the exponent in the function $(x^6 + 9)$?
4. How would the result change if we had a different expression inside the parentheses (e.g., $x^4 + 9$)?
5. What would happen if the exponent on $x^5$ was changed?

Tip: When using substitution in integrals, always try to match the derivative of the inside function to a part of the integrand for a simpler transformation.