91Ó°ÊÓ

When tackling the problem of finding an antiderivative—also known as solving an indefinite integral—it's important to be familiar with various integration techniques. In this particular exercise, the function we need to integrate is \(-2 \cos t\). Integration techniques involve identifying patterns and employing specific strategies to find the antiderivative. For instance:

• **Basic Integration Rules**: For basic functions, memorize common integrals like \( \int \cos t \, dt = \sin t + C \).
• **Constant Multiple Rule**: If a function is multiplied by a constant, such as \( -2 \), you can take the constant out of the integral: \( \int a \cdot f(t) \, dt = a \int f(t) \, dt \).

Applying these techniques to our exercise, we recognize that the integral of \(-2 \cos t\) involves the antiderivative of \(\cos t\), multiplied by \(-2\). This yields the antiderivative as \(-2 \sin t + C\), where \(C\) is a constant of integration.

When finding the most general antiderivative, the constant of integration, denoted usually as \(C\), plays a crucial role. This constant accounts for the family of functions in the solution. Each different value of \(C\) represents a different antiderivative.

Why do we add this constant? Because integration is the inverse process of differentiation. Differentiating any constant results in zero, which means that during integration, we lose any specific constant that might have been part of the original function.

In our example \(-2 \cos t\), the antiderivative solution is \(-2 \sin t + C\). By including \(C\), we ensure that our result includes every possible function that, when differentiated, yields the original \(-2 \cos t\). Essentially, this provides us with the complete set of solutions.

Checking your solution through differentiation is an essential step in integrating a function. This not only verifies your work but also reinforces the connection between these fundamental concepts.

To check if \(-2 \sin t + C\) is the correct antiderivative of \(-2 \cos t\), differentiate it with respect to \(t\). The derivative of \(-2 \sin t\) is obtained as follows:

• The derivative of \(-2 \sin t\) is \(-2 \cos t\), due to the derivative rule \( \frac{d}{dt} \sin t = \cos t \).
• The derivative of the constant \(C\) is zero.

Therefore, the differentiation of \(-2 \sin t + C\) is exactly \(-2 \cos t\), proving that our original integration was executed correctly. This method of checking highlights the harmonious relationship between integration and differentiation, confirming the accuracy of your solution.