What are the steps to integrate sin x and what is the result of the integration?

Integrating the function sin(x) is a fundamental concept in calculus. The integral of sin(x) is often one of the first integrals learned because it provides essential insights into trigonometric functions and their properties regarding area under curves. Here’s a detailed breakdown of the steps involved in this integration:

1. Understand the Integral:
   The indefinite integral of a function can be thought of as the reverse process of differentiation. For our case, we want to find the function whose derivative is sin(x).
2. Recall the Derivative:
   One useful fact is that the derivative of -cos(x) is sin(x):
• frac{d}{dx} [-cos(x)] = sin(x)
3. Set Up the Integral:
   Based on the above fact, we can express the integral of sin(x) as:
• ∫ sin(x) dx = -cos(x) + C
4. Apply the Constant of Integration:
   The C in our formula represents the constant of integration, which is important because indefinite integrals yield a family of functions that differ by a constant. This gives us the final result.
5. Summarize the Result:
   Thus, the integral of sin(x) is:
   ∫ sin(x) dx = -cos(x) + C

In summary, upon integrating sin(x), you arrive at -cos(x) + C, where C is the constant of integration. This result not only plays a crucial role in many applications across physics and engineering but also bridges the understanding of how trigonometric functions behave in various mathematical contexts.