What is the integral of cos(x) dx?

The integral of cos(x) with respect to x is indeed sin(x) + C, where C is the constant of integration. This result can be understood by recognizing that the derivative of sin(x) is cos(x). Therefore, when we are finding the indefinite integral of cos(x), we are essentially determining which function, when differentiated, will yield cos(x).

The process is as follows:

1. Recall the fact that differentiation and integration are inverse operations.

2. We know that the derivative of sin(x) is cos(x).

3. Consequently, when we reverse this operation (i.e., integrate), we arrive at sin(x) + C.

This constant C is added to account for the family of antiderivatives since the derivative of a constant is zero, thus allowing for infinitely many functions that can yield the same derivative.

This understanding leads us to the conclusion that the integral of cos(x) is sin(x) + C. Other options do not hold true in the context of basic trigonometric integrals, as they reflect either other incorrect antiderivatives of cos(x) or functions which have different derivatives.