Problem 59 - University Calculus: Early Transcendentals

Question

Find the most general antiderivative or indefinite integral. You may need to try a solution and then adjust your guess. Check your answers by differentiation. $$\int \frac{1+\cos 4 t}{2} d t$$

Step-by-Step Solution

Answer

The most general antiderivative is \( \frac{1}{2}t + \frac{1}{8}\sin 4t + C \).

Step 1: Split the Integral into Simple Terms

Rewrite the integral by splitting it into simpler terms: \[ \int \frac{1}{2} \, dt + \int \frac{\cos 4t}{2} \, dt. \] This separation helps in addressing each part individually.

Step 2: Integrate the Constant Term

The first part of the integral is \( \int \frac{1}{2} \, dt \). The antiderivative of a constant \( c \) is \( c \cdot t \). Therefore, the antiderivative for this term is: \[ \frac{1}{2}t. \]

Step 3: Integrate the Trigonometric Term

For the second part \( \int \frac{\cos 4t}{2} \, dt \), use the substitution rule to simplify integration: Set \( u = 4t \), then \( du = 4 \, dt \) or \( dt = \frac{1}{4} \, du \). Rewrite the integral as: \[ \frac{1}{2} \int \cos u \cdot \frac{1}{4} \, du = \frac{1}{8} \int \cos u \, du. \] The antiderivative of \( \cos u \) is \( \sin u \). Thus, the integral becomes: \[ \frac{1}{8} \sin u = \frac{1}{8} \sin 4t. \]

Step 4: Combine and Include the Constant of Integration

Combine the results from Steps 2 and 3 to present the most general antiderivative: \[ \frac{1}{2}t + \frac{1}{8}\sin 4t + C, \] where \( C \) is the constant of integration. This represents the family of all antiderivatives for the original function.

Step 5: Verify by Differentiation

To ensure correctness, differentiate the result: Differentiate \( \frac{1}{2}t + \frac{1}{8}\sin 4t + C \): - The derivative of \( \frac{1}{2}t \) is \( \frac{1}{2} \).- The derivative of \( \frac{1}{8}\sin 4t \) is \( \frac{1}{8} \cdot 4 \cos 4t = \frac{1}{2} \cos 4t \).- The derivative of \( C \) is 0 as it is a constant. The sum of these derivatives is: \[ \frac{1}{2} + \frac{1}{2} \cos 4t = \frac{1 + \cos 4t}{2}, \] which matches the original integrand, confirming our solution is correct.

Key Concepts

Antiderivative

The term "antiderivative" refers to a function that reverses the process of differentiation. It is a function whose derivative results in the original function we started with. Here, when we find the antiderivative of a given function, we are essentially looking to identify a function whose derivative will give us the original function.
The process of finding an antiderivative is called integration. Therefore, when we compute an indefinite integral of a function, we are determining its most general antiderivative. For example, the antiderivative of a constant function like \( rac{1}{2} \) is simply \( rac{1}{2}t \) because differentiating \( rac{1}{2}t \) yields \( rac{1}{2} \) again. This highlights how integration and differentiation are inverse operations.

Trigonometric Substitution

Trigonometric substitution is a technique used to simplify integrals that involve trigonometric functions. In our example, handling \( rac{1+ ext{cos } 4t}{2} \) involves splitting integrals and focusing on the trigonometric part separately.
The integral \( rac{1}{2} ext{cos } 4t \) requires us to use a substitution method. By substituting \( u = 4t \), we change the variable to free the integral from a cumbersome trigonometric relationship. This substitution results in \( du = 4 \, dt \), or \( dt = \frac{1}{4} \, du \). Now, the integral morphs into a simpler form: \( \frac{1}{8} \int \cos u \, du \), allowing us to perform integration with ease.
The antiderivative of \( \text{cos } u \) is \( \text{sin } u \), which reverses the differentiation process for cosine, providing the integral \( \frac{1}{8} \text{sin } 4t \). Thus, trigonometric substitution not only simplifies the integral but also provides insight into the antiderivative's function involving trigonometric expressions.

Differentiation

Differentiation is the process of finding the derivative of a function, which tells us how the function changes at any given point. In verifying our solution for the indefinite integral, differentiation is a critical step. After determining the general form of the antiderivative, we use differentiation to check if it aligns with the original function.
For the antiderivative \( \frac{1}{2}t + \frac{1}{8}\text{sin } 4t + C \), we must ensure it differentiates back to our original integrand \( \frac{1+ ext{cos } 4t}{2} \).

• The derivative of \( \frac{1}{2}t \) gives us \( \frac{1}{2} \).
• The derivative of \( \frac{1}{8}\text{sin } 4t \) becomes \( \frac{1}{8}\cdot 4 \cdot \text{cos } 4t = \frac{1}{2} \text{cos } 4t \).
• The derivative of \( C \), being a constant, is zero.

Adding these derivatives together, \( \frac{1}{2} + \frac{1}{2} \text{cos } 4t \), confirms the original expression \( \frac{1 + \text{cos } 4t}{2} \), verifying the accuracy of our antiderivative.

Constant of Integration

When finding an indefinite integral, it represents a family of possible antiderivatives, distinguished by what's referred to as the "constant of integration," often denoted as \( C \). Each function in this family results from adding a constant number to another antiderivative.
The constant of integration is crucial because when differentiating to verify an integral, any constant vanishes; hence, two functions differing by only a constant will still have the same derivative.
This is why, in any indefinite integral solution like \( \frac{1}{2}t + \frac{1}{8}\sin 4t + C \), the \( C \) encompasses the infinite set of vertical shifts of the function. It's a reminder that while our antiderivative represents one solution, there are indeed multiple solutions, each reflecting a unique member of this family of curves. Including the constant of integration ensures that we have accounted for all possible antiderivatives.