What is the result of the integral ∫ (sin(x) + cos(x)) dx?

What is the result of the integral ∫ (sin(x) + cos(x)) dx?

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

Answer: (A)

To determine the result of the integral of the function \( \sin(x) + \cos(x) \), we can integrate each component separately. The integral of \( \sin(x) \) is \( -\cos(x) \), and the integral of \( \cos(x) \) is \( \sin(x) \). Therefore, when we combine these results, the integral can be expressed as follows: \[ \int (\sin(x) + \cos(x)) \, dx = \int \sin(x) \, dx + \int \cos(x) \, dx = -\cos(x) + \sin(x) + C \] Here, \( C \) represents the constant of integration, which is included because the integral represents a family of antiderivatives. The correct answer reflects this derived expression perfectly, showcasing the individual integrals combined into one correct result. Thus, the correct answer is indeed expressed as \( -\cos(x) + \sin(x) + C \).

To determine the result of the integral of the function ( \sin(x) + \cos(x) ), we can integrate each component separately.

The integral of ( \sin(x) ) is ( -\cos(x) ), and the integral of ( \cos(x) ) is ( \sin(x) ). Therefore, when we combine these results, the integral can be expressed as follows:

[

\int (\sin(x) + \cos(x)) , dx = \int \sin(x) , dx + \int \cos(x) , dx = -\cos(x) + \sin(x) + C

]

Here, ( C ) represents the constant of integration, which is included because the integral represents a family of antiderivatives.

The correct answer reflects this derived expression perfectly, showcasing the individual integrals combined into one correct result. Thus, the correct answer is indeed expressed as ( -\cos(x) + \sin(x) + C ).