What is the integral of cos(x) dx?

The integral of cos(x) with respect to x is sin(x) + C, where C is the constant of integration. This result arises from the fundamental theorem of calculus, which states that the integral of a function gives the antiderivative.

When considering the function cos(x), you can recall its derivative. The derivative of sin(x) is cos(x). Therefore, when we seek the antiderivative, we find that the integral of cos(x) must lead us back to sin(x). To capture all possible antiderivatives, we add the constant of integration, C, since there are infinitely many functions that differ by a constant which all have the same derivative.

In the other options, we see formulations that do not represent the correct antiderivative of cos(x). The expression -sin(x) would represent the integral of -cos(x), and sin^2(x)/2 does not relate directly as an antiderivative of cos(x). Likewise, -cos(x) would be the integral of sin(x), which does not align with the integral of cos(x). Thus, the only correct representation of the integral of cos(x) is sin(x) + C.