Step-by-step Solution:

To solve the integral: \[ \int \frac{(x^5 - x)^{\frac{1}{5}}}{x^6} \, dx, \] we use substitution. Let: \[ u = 1 - \frac{1}{x^4}. \] Then, the derivative of \( u \) is: \[ du = \frac{4}{x^5} \, dx \quad \Rightarrow \quad \frac{du}{4} = \frac{1}{x^5} \, dx. \] Rewrite the integral in terms of \( u \): \[ \int \frac{(x^5 - x)^{\frac{1}{5}}}{x^6} \, dx = \int \left(x^5 - x\right)^{\frac{1}{5}} \cdot \frac{1}{x^6} \, dx. \] Factor \( x^5 - x \) as \( x(x^4 - 1) \), and rewrite the integrand: \[ \int \left(x^5 - x\right)^{\frac{1}{5}} \cdot \frac{1}{x^6} \, dx = \int \left(x(x^4 - 1)\right)^{\frac{1}{5}} \cdot \frac{1}{x^6} \, dx. \] Simplify: \[ = \int x^{\frac{1}{5}} (x^4 - 1)^{\frac{1}{5}} \cdot \frac{1}{x^6} \, dx = \int (x^4 - 1)^{\frac{1}{5}} \cdot x^{\frac{1}{5} - 6} \, dx. \] Further simplify the exponent of \( x \): \[ x^{\frac{1}{5} - 6} = x^{-\frac{29}{5}}. \] Now, substitute \( u = 1 - \frac{1}{x^4} \), and rewrite the integral: \[ \int (x^4 - 1)^{\frac{1}{5}} \cdot x^{-\frac{29}{5}} \, dx = \int u^{\frac{1}{5}} \cdot \frac{du}{4}. \] Integrate with respect to \( u \): \[ \frac{1}{4} \int u^{\frac{1}{5}} \, du = \frac{1}{4} \cdot \frac{u^{\frac{6}{5}}}{\frac{6}{5}} + C = \frac{5}{24} u^{\frac{6}{5}} + C. \] Substitute back \( u = 1 - \frac{1}{x^4} \): \[ \frac{5}{24} \left(1 - \frac{1}{x^4}\right)^{\frac{6}{5}} + C. \]