What is the integral of sin(x) dx?

What is the integral of sin(x) dx?

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

Answer: (A)

The integral of sin(x) with respect to x is indeed -cos(x) + C. This result arises from the fundamental relationship between differentiation and integration in calculus. To find the integral of sin(x), we can rely on the fact that differentiation of the cosine function gives us negative sine: d/dx [cos(x)] = -sin(x). Therefore, when we integrate sin(x), the function whose derivative we get as sin(x) must be -cos(x). The "+ C" represents the constant of integration, a necessary addition whenever we compute an indefinite integral, because integration can yield multiple functions differing only by a constant. This understanding is foundational in integral calculus. The other options do not correctly represent the integral of sin(x) due to misunderstandings about the relationship between sine and cosine functions or incorrect computations involving integrals.

The integral of sin(x) with respect to x is indeed -cos(x) + C. This result arises from the fundamental relationship between differentiation and integration in calculus.

To find the integral of sin(x), we can rely on the fact that differentiation of the cosine function gives us negative sine:

d/dx [cos(x)] = -sin(x).

Therefore, when we integrate sin(x), the function whose derivative we get as sin(x) must be -cos(x). The "+ C" represents the constant of integration, a necessary addition whenever we compute an indefinite integral, because integration can yield multiple functions differing only by a constant.

This understanding is foundational in integral calculus. The other options do not correctly represent the integral of sin(x) due to misunderstandings about the relationship between sine and cosine functions or incorrect computations involving integrals.