Problem 25 - Applied Calculus

Question

Find an antiderivative. $$ g(t)=5+\cos t $$

Step-by-Step Solution

Answer

The antiderivative of \( g(t) = 5 + \cos t \) is \( 5t + \sin t + C \).

Step 1: Understand the Function

The function given is \( g(t) = 5 + \cos t \). To find an antiderivative, integrate each term separately with respect to \( t \).

Step 2: Integrate the Constant Term

The first term is a constant \( 5 \). The antiderivative of a constant \( a \) with respect to \( t \) is \( at + C_1 \), where \( C_1 \) is a constant of integration. Here, integrate \( 5 \) to get \( 5t + C_1 \).

Step 3: Integrate the Trigonometric Term

The second term is \( \cos t \). The antiderivative of \( \cos t \) with respect to \( t \) is \( \sin t + C_2 \), where \( C_2 \) is another constant of integration. Integrate to get \( \sin t + C_2 \).

Step 4: Combine the Antiderivatives

Add the two antiderivatives obtained from the separate integrations: \( 5t + \sin t + C_1 + C_2 \). Instead of using two separate constants, combine them into a single constant \( C \). The general antiderivative is thus \( 5t + \sin t + C \).

Step 5: Finalize the Antiderivative

Conclude with the final expression for the antiderivative, combining all the elements derived. The antiderivative of \( g(t) = 5 + \cos t \) is \( 5t + \sin t + C \), where \( C \) is an arbitrary constant.

Key Concepts

Integration Techniques

Integration techniques are methods used to find antiderivatives, which are functions that reverse the process of differentiation. If you recall differentiation as breaking down functions into rates of change, integration builds functions from these rates. When faced with an expression like \( g(t) = 5 + \cos t \), each term can be integrated separately.

• Integrate Constants: For constant terms like 5, integration simply involves multiplying the constant by the variable, which in this case is \( t \). So, the antiderivative of 5 is \( 5t + C_1 \).
• Basic Functions: For more complex functions like trigonometric functions, basic formulas are used. For \( \cos t \), the antiderivative is \( \sin t + C_2 \).

Learning integration techniques allows students to handle a variety of functions, applying appropriate strategies to each component.

Trigonometric Integration

Trigonometric integration is a specific branch of integration that deals with trigonometric functions such as sine and cosine. These functions have well-known derivatives which can be reversed to find their antiderivatives more easily.

• The antiderivative of \( \cos t \) is \( \sin t \), based on the fundamental derivative relationship that the derivative of \( \sin t \) is \( \cos t \).
• Such integration allows us to find expressions that model oscillations and periodic behavior in real-world phenomena.

In the given example, knowing that the derivative of \( \sin t \) is \( \cos t \) lets us directly state that integrating \( \cos t \) yields \( \sin t + C_2 \). Understanding this relationship is key for solving problems that involve trigonometric formulas.

Constants of Integration

In integration, the constants of integration are crucial for representing the most general form of an antiderivative. When we integrate any function, we're essentially finding a family of functions that all differentiate to the original function.

• Each term in the antiderivative can have its own integration constant, but typically, these are combined into a single expression \( C \).
• The constant of integration accounts for the fact that there are infinitely many functions with the same derivative scattered through the vertical plane.

For instance, when integrating the components of \( g(t) = 5 + \cos t \), we initially have \( C_1 \) and \( C_2 \) for each term. These are combined into one term \( C \), simplifying the final antiderivative to \( 5t + \sin t + C \). This simplified form is elegant and practical for mathematical and real-world applications.