What is the integral of cos(x) dx?

The integral of cos(x) with respect to x is a fundamental result in calculus, particularly in the study of integrals. The process of finding the integral essentially reverses the process of differentiation. Since the derivative of sin(x) is cos(x), integrating cos(x) will yield sin(x).

Thus, we can express the integral of cos(x) as:

∫cos(x) dx = sin(x) + C,

where C represents the constant of integration. This captures all possible antiderivatives of cos(x), reflecting that there are infinitely many functions that vary by a constant, each of which has the same derivative of cos(x).

The constant C is necessary because, during differentiation, the constant term disappears, meaning any constant added to sin(x) would still differentiate to cos(x).

This understanding is fundamental for integrating trigonometric functions, making the choice of sin(x) + C the correct answer. The other choices do not correctly represent the integral of cos(x), as they either reflect other trigonometric functions or are incorrect in their differentiation results.