1. **State the problem:** We need to find the Jacobian of a transformation scaled by the determinant of the matrix $\begin{bmatrix} 2 & 6 & 5 \end{bmatrix}$ and then evaluate the integral of the vector function $x^2 + 6xy + 6z$. 2. **Jacobian and determinant:** The Jacobian matrix is the matrix of all first-order partial derivatives of a vector function. The determinant of a matrix $A$ is a scalar value that can scale volumes when transforming coordinates. 3. **Calculate the determinant:** The given matrix is a 1x3 vector, so its determinant is not defined. Possibly, the matrix intended is a 3x3 matrix or the determinant refers to a scalar value from a square matrix. Since only one row is given, we cannot compute the determinant. 4. **Integral of the vector function:** Assuming the integral is with respect to $x$, and treating $y$ and $z$ as constants, the integral is: $$\int (x^2 + 6xy + 6z) \, dx = \int x^2 \, dx + \int 6xy \, dx + \int 6z \, dx$$ 5. **Compute each integral:** - $\int x^2 \, dx = \frac{x^3}{3} + C_1$ - $\int 6xy \, dx = 6y \int x \, dx = 6y \frac{x^2}{2} + C_2 = 3yx^2 + C_2$ - $\int 6z \, dx = 6z x + C_3$ 6. **Combine results:** $$\frac{x^3}{3} + 3yx^2 + 6zx + C$$ where $C = C_1 + C_2 + C_3$ is the constant of integration. 7. **Summary:** Since the determinant of the given vector is undefined, we focus on the integral of the vector function with respect to $x$. The integral is: $$\frac{x^3}{3} + 3yx^2 + 6zx + C$$