$\int \sqrt[5]{r^{4}}$

Step 1 ：The integral of a function is the area under the curve of the function. In this case, we are asked to find the integral of \(\sqrt[5]{r^{4}}\), which is the same as \(r^{4/5}\).

Step 2 ：To solve this, we can use the power rule for integration, which states that the integral of \(x^n\) is \(\frac{1}{n+1}x^{n+1}\).

Step 3 ：Applying the power rule to our function, we get \(\frac{5}{9}r^{9/5}\).

Step 4 ：Final Answer: \(\boxed{\frac{5}{9}r^{9/5}}\)