\[ \text { Use a change of variables or the table to evaluate the following indefinite integral. } \] \[ x^{4}\left(x^{5}+2\right)^{3} d x \] \( \int x^{4}\left(x^{5}+2\right)^{3} d x=\square \)

To evaluate the integral \(\int x^{4}\left(x^{5}+2\right)^{3} d x\), we can use the substitution method. Let’s set \(u = x^5 + 2\). Then, the derivative of \(u\) with respect to \(x\) is \(du = 5x^4 dx\), or rearranged, \(dx = \frac{du}{5x^4}\). Now substituting, we notice that \(x^4 dx\) is needed for our integral, which means: \[ x^4 dx = \frac{1}{5} du. \] Now we can change the integral: \[ \int x^{4}\left(x^{5}+2\right)^{3} dx = \int \left(u\right)^{3} \left(\frac{1}{5} du\right) = \frac{1}{5} \int u^{3} du. \] Next, we evaluate the integral of \(u^{3}\): \[ \frac{1}{5} \left(\frac{u^{4}}{4}\right) + C = \frac{1}{20} u^{4} + C. \] Substituting back \(u = x^5 + 2\), we get: \[ \int x^{4}\left(x^{5}+2\right)^{3} d x = \frac{1}{20} (x^5 + 2)^{4} + C. \] So the final answer is: \[ \int x^{4}\left(x^{5}+2\right)^{3} d x = \frac{1}{20} (x^5 + 2)^{4} + C. \]