Determine the integral ∫ cos(x) dx. What do you obtain?

Determine the integral ∫ cos(x) dx. What do you obtain?

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

Answer: (A)

The integral of cos(x) with respect to x can be derived using basic integration techniques. Specifically, the integral ∫ cos(x) dx can be calculated by recognizing that the derivative of sin(x) is equal to cos(x). This relationship is fundamental in calculus. When integrating cos(x), we effectively reverse the differentiation process. As a result, the integral yields sin(x) as the primary function, and since we are dealing with indefinite integrals, we must also include the constant of integration, C. Hence, the complete result is sin(x) + C. This understanding stems from the fundamental theorem of calculus, which establishes a direct link between differentiation and integration. Therefore, when one integrates cos(x), one rightfully arrives at the conclusion that sin(x) is the antiderivative of cos(x), leading to the answer being sin(x) + C. The other options suggest different functions or their negations, which do not relate to the integrals of cosine. Introducing them does not conform to the basic principles of integration with respect to trigonometric functions.

The integral of cos(x) with respect to x can be derived using basic integration techniques. Specifically, the integral ∫ cos(x) dx can be calculated by recognizing that the derivative of sin(x) is equal to cos(x). This relationship is fundamental in calculus.

When integrating cos(x), we effectively reverse the differentiation process. As a result, the integral yields sin(x) as the primary function, and since we are dealing with indefinite integrals, we must also include the constant of integration, C. Hence, the complete result is sin(x) + C.

This understanding stems from the fundamental theorem of calculus, which establishes a direct link between differentiation and integration. Therefore, when one integrates cos(x), one rightfully arrives at the conclusion that sin(x) is the antiderivative of cos(x), leading to the answer being sin(x) + C.

The other options suggest different functions or their negations, which do not relate to the integrals of cosine. Introducing them does not conform to the basic principles of integration with respect to trigonometric functions.