[tex]1. Simplify the integral by factoring out the common factor of \(x^{2}\) inside the square root:
\(\int x^{2} \sqrt{2x^{3} + 7} dx\)
2. Use substitution method: Let \(u = 2x^{3} + 7\)
Then, differentiate both sides with respect to x to find \(du\): \(du = 6x^{2} dx\)
3. Rearrange the equation to solve for \(dx\): \(dx = \frac{du}{6x^{2}}\)
4. Substitute \(u\) and \(dx\) back into the integral:
\(\frac{1}{6} \int \sqrt{u} du\)
5. Integrate \(\int \sqrt{u} du\) with respect to \(u\):
\(\frac{1}{6} \cdot \frac{2}{3} u^{\frac{3}{2}} + C\)
\: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: 6. Simplify \: the \: integrated \: form \: to \: get \: the \: final solution: \: \: \:
\(\frac{1}{9} (2x^{3} + 7)^{\frac{3}{2}} + C\)[/tex]
