91Ó°ÊÓ

Integration is a key concept in calculus and a crucial tool in mathematics. It allows us to find quantities like areas under curves, among other physical and geometrical applications. An indefinite integral, often referred to simply as an antiderivative, of a function is a function whose derivative is the original function.
In the given exercise, we're asked to find the indefinite integral of \( \cos x - \sin x \). We tackle this by applying basic integration rules for each term separately. Breaking it into simpler parts is always a good strategy.
First, we split the integral into two parts: \[ \int (\cos x - \sin x) \, dx = \int \cos x \, dx - \int \sin x \, dx \]
This approach helps us manage and solve each component efficiently by applying standard integration formulas.
Remember, when integrating, constants of integration might appear, and combining them into a single constant simplifies our final answer.

Trigonometric functions like \cos x\ and \sin x\ frequently appear in calculus and are pivotal in solving many problems. For integrated functions, remember:

• The integral of \cos x \, dx \ is \sin x + C_1\
• The integral of \sin x \, dx \ is \-\cos x + C_2\

Trigonometric integrals require recognizing and applying these fundamental antiderivatives. For instance, in the given exercise, when you separately integrate each term:

• \[ \int \cos x \, dx = \sin x + C_1 \]
• \[ \int \sin x \, dx = -\cos x + C_2 \]

Trigonometric functions often exhibit periodic behavior and symmetry, which can be useful for simplifying and solving integrals.

Effective calculus problem-solving involves breaking down complex problems into more manageable steps. Here’s a simple guide:

• Identify the Problem: Recognize the type of integral or function you're dealing with. In this case, it's an indefinite integral involving trigonometric functions.
• Simplify and Separate: Break the integral into simpler parts, as seen by separating \[ \cos x - \sin x \] into \[ \int \cos x \, dx \] and \[ \int \sin x \, dx \].
• Use Basic Integration Rules: Apply basic rules and known antiderivatives for each term.
• Combine Results: After integrating each part, combine them while taking care of constants of integration, reducing them to a single constant \[ C \].

So, the solution to the exercise involves following these steps: Split, integrate each part separately, and combine the results.
Finally, the answer to \ \int (\cos x - \sin x) \, dx \ \ is \[ \sin x - \cos x + C \].