Final answer:
The antiderivative of the function f(x) = cos(2x) - 1 is (1/2)sin(2x) - x + C, where C is an arbitrary constant that represents the family of antiderivatives.
Step-by-step explanation:
The question asks to find the antiderivative of the function f(x) = cos(2x) - 1. To find the antiderivative, we need to integrate the function concerning x. Let's integrate each component separately.
- For cos(2x), the antiderivative is (1/2)sin(2x), because the derivative of sin(2x) is 2cos(2x), and we want to reverse this process.
- For the constant -1, the antiderivative is -x, since the derivative of -x is -1.
So, the antiderivative of f(x) is (1/2)sin(2x) - x, plus an arbitrary constant C, since antiderivatives are determined up to an additive constant.
Therefore, the answer is ∫ (cos(2x) - 1) dx = (1/2)sin(2x) - x + C.