Find $\int\left(4 x^{5}+2 x^{6}\right) d x$

Step 1 ：Split the integral into two parts: \(\int 4x^5 dx\) and \(\int 2x^6 dx\).

Step 2 ：Find the integral of each part separately using the rule \(\int x^n dx = \frac{1}{n+1}x^{n+1}\).

Step 3 ：For the first part, \(\int 4x^5 dx = 4 \int x^5 dx = 4 \cdot \frac{1}{5+1}x^{5+1} = \frac{4}{6}x^6\).

Step 4 ：For the second part, \(\int 2x^6 dx = 2 \int x^6 dx = 2 \cdot \frac{1}{6+1}x^{6+1} = \frac{2}{7}x^7\).

Step 5 ：Add the results of the two integrals together to get the final answer: \(\frac{2}{7}x^7 + \frac{2}{3}x^6 + C\), where \(C\) is the constant of integration.

Step 6 ：\(\boxed{\frac{2}{7}x^7 + \frac{2}{3}x^6 + C}\) is the final answer.