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DETAILS SCALC9 4.4.035.MI.SA.
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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: \(\boxed{7.41269841269841}\)