Find the integral. \( \int\left(x^{4}-2 x^{3}\right)^{4}\left(4 x^{3}-6 x^{2}\right) d x \) A. \( \frac{1}{5}\left(x^{4}-2 x^{3}\right)^{5}+C \) B. \( \frac{1}{4}\left(x^{4}-2 x^{3}\right)^{4}+C \) C. \( \left(x^{4}-2 x^{3}\right)^{5}+C \) D. \( 4 x^{3}-6 x^{2}+C \)

To solve the integral \( \int\left(x^{4}-2 x^{3}\right)^{4}\left(4 x^{3}-6 x^{2}\right) d x \), we can use substitution. Let \( u = x^{4} - 2x^{3} \). Then, the derivative \( \frac{du}{dx} = 4x^{3} - 6x^{2} \), so \( du = (4x^{3} - 6x^{2}) dx \). Substituting into the integral gives us \( \int u^{4} du \), which integrates to \( \frac{1}{5} u^{5} + C \). Re-substituting for \( u \), we get \( \frac{1}{5} (x^{4} - 2x^{3})^{5} + C \). Thus, the correct answer is (A) \( \frac{1}{5}\left(x^{4}-2 x^{3}\right)^{5}+C \). Did you know the method of substitution, also known as “u-substitution,” is like a magic trick in calculus? It helps simplify complex integrals into more manageable forms. By cleverly changing variables, you can often transform an intimidating expression into something much simpler to integrate! Also, keep in mind that mastering integration involves recognizing patterns. The more you practice, the more you'll start to see integral forms that can be simplified using tricks like substitution, integration by parts, or trigonometric identities. Don’t worry if you make mistakes; they're just stepping stones to becoming an integration pro!