Question

Find the general antiderivative for the function below; assume we have chosen an interval i on which the function is continuous. the function may require an algebraic manipulation first. (use c for the constant of integration. remember to use absolute values where appropriate.) f(x) = (4 x )2 x?

Answer 1

The general antiderivative of the function is found by integrating 4x^3, which yields x^4 + C, where C is the constant of integration.

Step-by-step explanation:

To find the general antiderivative for the given function f(x) = (4x)^2x, we need to interpret the function correctly and determine the antiderivative. The function as written is unclear, but we might assume it is meant to represent either (4x^2)x or 4x^(2x). However, the latter would involve the antiderivative of an exponential function with a variable exponent, which is not elementary. Assuming the former, which is 4x^3, we can integrate:

\(\int 4x^3 \, dx = x^4 + C\)

where \(C\) is the constant of integration. When integrating functions, it's important to consider if there's a need for absolute values, but in this case, since any real number raised to the fourth power is positive, absolute values are not required.