7. (16 points) Evaluate by substitution: \[ \int_{0}^{4} \frac{6 x^{3}}{\sqrt{x^{2}+9}} d x \] (a) (2pts) Identify the inner function \[ u= \] (b) (2pts) Derive both sides of (a) to get \[ d u= \] (c) (2pts) Isolate $d x$ \[ d x= \] (d) (10pts) Proceed to substitution and integrating.

Step 1 ：The inner function is \(u = x^{2} + 9\).

Step 2 ：Deriving both sides of the equation \(u = x^{2} + 9\) with respect to \(x\) gives \(du = 2x dx\).

Step 3 ：Isolating \(dx\) gives \(dx = \frac{du}{2x}\).

Step 4 ：Substitute \(u = x^{2} + 9\) and \(dx = \frac{du}{2x}\) into the integral, we get \(\int_{0}^{4} \frac{6 x^{3}}{\sqrt{u}} \cdot \frac{du}{2x} = 3\int_{0}^{4} \frac{x^{2}}{\sqrt{u}} du\).

Step 5 ：Substitute \(u = x^{2} + 9\) into the integral, we get \(3\int_{9}^{25} \frac{u - 9}{\sqrt{u}} du\).

Step 6 ：Split the integral into two parts, we get \(3\int_{9}^{25} \frac{u}{\sqrt{u}} du - 3\int_{9}^{25} \frac{9}{\sqrt{u}} du\).

Step 7 ：Simplify the integral, we get \(3\int_{9}^{25} u^{\frac{1}{2}} du - 27\int_{9}^{25} u^{-\frac{1}{2}} du\).

Step 8 ：Integrate the integral, we get \(3[\frac{2}{3}u^{\frac{3}{2}}]_{9}^{25} - 27[2u^{\frac{1}{2}}]_{9}^{25}\).

Step 9 ：Calculate the integral, we get \(2[25^{\frac{3}{2}} - 9^{\frac{3}{2}}] - 54[25^{\frac{1}{2}} - 9^{\frac{1}{2}}]\).

Step 10 ：\boxed{2[25^{\frac{3}{2}} - 9^{\frac{3}{2}}] - 54[25^{\frac{1}{2}} - 9^{\frac{1}{2}}]} is the final answer.