Problem
2. [-/1 Points] DETAILS SCALC9 4.5.003.MI. MY NOTES ASK YOUR TEACHER PRACTICE ANOTHER Evaluate the integral by making the given substitution. (Use $C$ for the constant of integration.) \[ \int x^{2} \sqrt{x^{3}+13} d x, \quad u=x^{3}+13 \] Need Help? Read It Watch it Master It
Solution
Step 1 ：Given the integral \(\int x^{2} \sqrt{x^{3}+13} dx\) and the substitution \(u=x^{3}+13\).
Step 2 ：Differentiate \(u\) with respect to \(x\) to get \(du\), \(du = 3x^{2} dx\).
Step 3 ：Rearrange the equation to solve for \(dx\), \(dx = \frac{du}{3x^{2}}\).
Step 4 ：Substitute \(u\) and \(dx\) into the integral, \(\int x^{2} \sqrt{u} \frac{du}{3x^{2}}\).
Step 5 ：Simplify the integral to \(\frac{1}{3} \int \sqrt{u} du\).
Step 6 ：Evaluate the integral, \(\frac{1}{3} \frac{2}{3} u^{\frac{3}{2}} + C\).
Step 7 ：Substitute \(u\) back into the equation, \(\frac{2}{9} (x^{3}+13)^{\frac{3}{2}} + C\).
Step 8 ：So, the solution to the integral is \(\boxed{\frac{2}{9} (x^{3}+13)^{\frac{3}{2}} + C}\).
