Determine thie integral O. \( \frac{1}{12} \sqrt{(8 x+4)^{3}}+c \) b. \( \frac{1}{12} \sqrt[3]{(8 x+4)^{2}}+c \) O. c. \( 25(2 x+3) \) d. \( \frac{1}{12} \sqrt{(8 x+4)}+c \) O e. None of the given options is correct.

Let’s treat this as a fun exploration of integration! The integral you're working with often comes from polynomial expressions, and can be solved through substitution or direct application of formulas. In the case of polynomials, consider the power rule and the importance of adding the constant of integration \( c \). It’s essential to remember that every indefinite integral generates a family of functions, which is why including \( c \) is fundamental! Also, when it comes to practice, tackling integration problems can often lead to common mistakes, such as integrating without considering the constant or misapplying the power rule. Always double-check your differentiation skills as they can guide you back from errors made during integration!