\( \frac{\text { Integration using u-substitution }}{\text { 9. } \int \frac{15 x^{4}-4 x^{2}}{\sqrt{3 x^{5}-x^{3}}} d x} \)

To tackle the integral \( \int \frac{15 x^{4}-4 x^{2}}{\sqrt{3 x^{5}-x^{3}}} d x \), we can use u-substitution. Let’s set \( u = 3 x^5 - x^3 \). Then, we differentiate to find \( du = (15 x^4 - 3 x^2) dx \) or rearranging, \( \frac{du}{d x} = 15 x^4 - 3 x^2 \). Now we can rewrite \( 15 x^4 - 4 x^2 \) in terms of \( du \). Notice that \( 15 x^4 - 4 x^2 = (15 x^4 - 3 x^2) - (4 x^2 - 3 x^2) = (15 x^4 - 3 x^2) - x^2 \). Thus, we rewrite the integral, simplifying our expression which makes the substitution much cleaner: \[ \int \frac{(du + x^2) }{\sqrt{u}} \] Now we can separately integrate those terms to yield a final answer! As for selecting two additional aspects of integration with u-substitution: Integration using u-substitution is a clever way to simplify complex integrals by substituting a function to reveal simpler forms. Imagine like switching gears in a bicycle; when the slope gets steep, shifting gears helps to make pedaling easier! The same concept applies here – choose the right \( u \) and the integral can become straight downhill! Another fun fact, u-substitution is often mistakenly skipped, leading to overly complicated expressions or incorrect answers! Make sure to identify a proper substitution by looking for composite functions or nested functions – don’t just rush into integration! It's a treasure hunt; choosing the right \( u \) can often lead to finding the integral's hidden gems.