What does the integral of sin(x) dx yield?

What does the integral of sin(x) dx yield?

• A. -cos(x) + C
• B. cos(x) + C
• C. -sin(x) + C
• D. sin(x) + C

Answer: (A)

The integral of sin(x) with respect to x can be determined through the fundamental concept of antiderivatives. When integrating sin(x) dx, we are essentially looking for a function whose derivative is sin(x). The derivative of -cos(x) is indeed sin(x), since the cosine function has a well-known derivative that behaves as follows: differentiating cos(x) gives us -sin(x). Thus, when we consider the negative sign in front of cos(x), we find that the derivative of -cos(x) is sin(x), confirming that -cos(x) is the correct antiderivative. Moreover, the "+ C" is added to reflect the constant of integration, which represents the fact that there are infinitely many antiderivatives differing by a constant. Thus, -cos(x) + C accurately describes the indefinite integral of sin(x) dx. This reasoning supports why the result is -cos(x) + C.

The integral of sin(x) with respect to x can be determined through the fundamental concept of antiderivatives. When integrating sin(x) dx, we are essentially looking for a function whose derivative is sin(x).

The derivative of -cos(x) is indeed sin(x), since the cosine function has a well-known derivative that behaves as follows: differentiating cos(x) gives us -sin(x). Thus, when we consider the negative sign in front of cos(x), we find that the derivative of -cos(x) is sin(x), confirming that -cos(x) is the correct antiderivative.

Moreover, the "+ C" is added to reflect the constant of integration, which represents the fact that there are infinitely many antiderivatives differing by a constant. Thus, -cos(x) + C accurately describes the indefinite integral of sin(x) dx. This reasoning supports why the result is -cos(x) + C.