\( \int \frac { x ^ { 2 } } { \sqrt { 8 x - x ^ { 2 } } } d x \)

To solve the integral \( \int \frac{x^2}{\sqrt{8x - x^2}} \, dx \), we can start by simplifying the expression under the square root. Rewrite it as \( \sqrt{- (x^2 - 8x)} = \sqrt{- (x(x - 8))} \). Next, let's use a trigonometric substitution. We can set \( x = 4 + 4\sin(\theta) \), transforming the integral to accommodate the parameterized limits. This utilizes the identity \( \sin^2(\theta) + \cos^2(\theta) = 1 \), allowing us to express the square root in terms of \( \theta \). After performing the substitution and simplifying, we can evaluate the resulting integral. To find exact values, completion of the square or substitution back into the integral will help achieve a solution. Conclude by substituting back to original variables and evaluating bounds if definite integral conditions apply. Happy integrating!