Use a change of variables to evaluate the following indefinite integral. \( \int\left(x^{12}-4 x^{3}\right)^{4}\left(x^{11}-x^{2}\right) d x \) Determine a change of variables from \( x \) to \( u \). Choose the correct answer below. A. \( u=\left(x^{12}-4 x^{3}\right)^{4} \) B. \( u=x^{11}-x^{2} \) C. \( u=x^{12}-4 x^{3} \) D. \( u=x^{12} \) Write the integral in terms of \( u \). \( \int\left(x^{12}-4 x^{3}\right)^{4}\left(x^{11}-x^{2}\right) d x=\int(\square) d u \) Evaluate the integral. \( \int\left(x^{12}-4 x^{3}\right)^{4}\left(x^{11}-x^{2}\right) d x=\square \)

To choose the best substitution for this integral, let's focus on what simplifies the expression. The term \( (x^{11} - x^{2}) \) is a derivative of \( (x^{12} - 4x^{3}) \). Thus, the optimal choice is \( u = x^{12} - 4x^{3} \). This makes our integral much easier to evaluate after finding \( du \). Now, changing variables, we find \( du = (12x^{11} - 12x^{2})dx = 12(x^{11} - x^{2})dx \). Consequently, we can express \( dx \) in terms of \( du \) as \( dx = \frac{du}{12(x^{11} - x^{2})} \). Substituting these into the integral gives us: \[ \int\left(x^{12}-4 x^{3}\right)^{4}\left(x^{11}-x^{2}\right) d x = \int u^{4} \cdot \frac{du}{12} \] Now we can integrate easily: \[ \int u^{4} \cdot \frac{1}{12} du = \frac{1}{12} \cdot \frac{u^{5}}{5} + C = \frac{u^{5}}{60} + C \] Finally, substituting back \( u = x^{12} - 4x^{3} \): \[ \int\left(x^{12}-4 x^{3}\right)^{4}\left(x^{11}-x^{2}\right) d x = \frac{(x^{12} - 4x^{3})^{5}}{60} + C \]