3. $[-/ 4$ Points] DETAILS SCALC9 4.5.003.MI.SA. MY NOTES ASK YOUR TEACHER PRACTICE ANOTHER This question has several parts that must be completed sequentially. If you skip a part of the question, you will not receive any points for the skipped part, and you will not be able to come back to the skipped part. Tutorial Exercise Evaluate the integral by making the given substitution. \[ \int x^{2} \sqrt{x^{3}+24} d x, \quad u=x^{3}+24 \] Step 1 We know that if $u=f(x)$, then $d u=f^{\prime}(x) d x$. Therefore, if $u=x^{3}+24$, then $d u=$ $d x$ Need Help? Read It Submit Answer

Step 1 ：Given the integral \(\int x^{2} \sqrt{x^{3}+24} dx\) and the substitution \(u=x^{3}+24\).

Step 2 ：Differentiate \(u=x^{3}+24\) with respect to \(x\) to find \(du\). This gives \(du=3x^{2}dx\).

Step 3 ：Rearrange the equation \(du=3x^{2}dx\) to solve for \(dx\). This gives \(dx=\frac{du}{3x^{2}}\).

Step 4 ：Substitute \(u\) and \(dx\) into the integral. This gives \(\int x^{2} \sqrt{u} \frac{du}{3x^{2}}\).

Step 5 ：Simplify the integral to \(\int \frac{1}{3} \sqrt{u} du\).

Step 6 ：Evaluate the integral to find \(\frac{2}{9} u^{\frac{3}{2}} + C\), where \(C\) is the constant of integration.

Step 7 ：Substitute \(x^{3}+24\) back in for \(u\) to find the final answer. This gives \(\boxed{\frac{2}{9} (x^{3}+24)^{\frac{3}{2}} + C}\).