Evaluate the indefinite integral.
\[
\int \frac{x^{2}}{\sqrt{1-x^{6}}} d x
\]
Answer
Final Answer: \(\boxed{\frac{x \Gamma\left(\frac{1}{6}\right) \, _2F_1\left(\frac{1}{6}, \frac{1}{2}; \frac{7}{6}; x^{6}\right)}{18 \Gamma\left(\frac{7}{6}\right)}}\)
Step 1 ：Let's start by making a substitution. Let \(u = x^3\), then \(du = 3x^2 dx\).
Step 2 ：This simplifies the integral to \(\int \frac{1}{3} \frac{du}{\sqrt{1-u^2}}\), which is a standard integral that can be solved using trigonometric substitution.
Step 3 ：The integral has been evaluated and the result is in terms of special functions, specifically the gamma function and the hypergeometric function.
Step 4 ：Final Answer: \(\boxed{\frac{x \Gamma\left(\frac{1}{6}\right) \, _2F_1\left(\frac{1}{6}, \frac{1}{2}; \frac{7}{6}; x^{6}\right)}{18 \Gamma\left(\frac{7}{6}\right)}}\)
