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

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.

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.