In the expression -cos(x) + C, what does C represent?

• A. The integration constant
• B. The final value
• C. The initial condition
• D. The amplitude

Answer: (A)

The main idea here is the constant of integration that appears when you find an antiderivative. When you integrate a function, you don’t just get a single antiderivative—you get a family of them that differ by a constant, because the derivative of any constant is zero. So -cos(x) + C is an antiderivative of sin(x), and C represents this freedom to add any constant.

If you know a specific point on the curve, you can determine C. For example, if at x = x0 the value is y0, then y0 = -cos(x0) + C, which gives C = y0 + cos(x0).

This isn’t a final value or an initial condition itself, though an initial condition can be used to fix C. It also isn’t an amplitude (which would scale the cosine term); it’s an additive constant that shifts the entire curve up or down without changing its slope.