What is the integral of sin(x) dx?

The integral of sin(x) with respect to x is -cos(x) + C, where C represents the constant of integration. This result can be understood through the concept of antiderivatives. When you differentiate -cos(x), you obtain sin(x). Therefore, the integral of sin(x) must yield a function which, when differentiated, results in sin(x).

To clarify further, the process of integration seeks to find a function whose derivative gives the integrand, in this case, sin(x). The negative sign in front of cos(x) is crucial because the derivative of cos(x) is -sin(x), confirming that -cos(x) is indeed the correct antiderivative for sin(x).

The inclusion of the constant C is standard in indefinite integrals to account for all possible vertical shifts of the antiderivative. Hence, -cos(x) + C is the general solution for the integral of sin(x).