15. $[-19$ Points $]$ DETAILS SCALC9 4.4.035.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. \[ \int_{0}^{1} x(9 \sqrt[3]{x}+8 \sqrt[4]{x}) d x \] Step 1

Step 1 ：Given the integral \(\int_{0}^{1} x(9 \sqrt[3]{x}+8 \sqrt[4]{x}) d x\)

Step 2 ：Distribute the x inside the parentheses to get \(\int_{0}^{1} (9x^{4/3}+8x^{5/4}) d x\)

Step 3 ：Separate the integral into two parts: \(\int_{0}^{1} 9x^{4/3} dx + \int_{0}^{1} 8x^{5/4} dx\)

Step 4 ：Calculate the integral of each part separately

Step 5 ：The integral of \(9x^{4/3}\) from 0 to 1 is \(\frac{27}{7}\)

Step 6 ：The integral of \(8x^{5/4}\) from 0 to 1 is \(\frac{32}{9}\)

Step 7 ：Add the results of the two integrals to get the final answer

Step 8 ：\(\frac{27}{7} + \frac{32}{9} = 7.41269841269841\)

Step 9 ：Final Answer: \(\boxed{7.41269841269841}\)