Evaluate the integral ∫ (sin(x) + cos(x)) dx.

Evaluate the integral ∫ (sin(x) + cos(x)) dx.

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

Answer: (C)

To evaluate the integral ∫ (sin(x) + cos(x)) dx, we can break it down into two separate integrals: 1. The integral of sin(x) dx 2. The integral of cos(x) dx The integral of sin(x) with respect to x is -cos(x), and the integral of cos(x) with respect to x is sin(x). Therefore, we can write the combined integral as follows: ∫ (sin(x) + cos(x)) dx = ∫ sin(x) dx + ∫ cos(x) dx = -cos(x) + sin(x) + C Here, C represents the constant of integration, which is included in the final answer as indefinite integrals do not yield a unique solution but rather a family of functions that differ by a constant. This matches option C: -cos(x) + sin(x) + C, which reflects the correct evaluation of the integral. The order of the terms does not change the validity of the expression since addition is commutative. Thus, option C is indeed the correct choice.

To evaluate the integral ∫ (sin(x) + cos(x)) dx, we can break it down into two separate integrals:

1. The integral of sin(x) dx

2. The integral of cos(x) dx

The integral of sin(x) with respect to x is -cos(x), and the integral of cos(x) with respect to x is sin(x). Therefore, we can write the combined integral as follows:

∫ (sin(x) + cos(x)) dx = ∫ sin(x) dx + ∫ cos(x) dx = -cos(x) + sin(x) + C

Here, C represents the constant of integration, which is included in the final answer as indefinite integrals do not yield a unique solution but rather a family of functions that differ by a constant.

This matches option C: -cos(x) + sin(x) + C, which reflects the correct evaluation of the integral. The order of the terms does not change the validity of the expression since addition is commutative. Thus, option C is indeed the correct choice.