Problem 27 - Calculus Early Transcendentals: Pearson New International Edition

Question

Evaluate the indicated indefinite integrals. $$ \int(\sin \theta-\cos \theta) d \theta $$

Step-by-Step Solution

Answer

\( -\cos \theta - \sin \theta + C \)

Step 1: Identify the Components

The indefinite integral we need to evaluate is \( \int (\sin \theta - \cos \theta) \, d\theta \). This expression is a sum of two functions: \( \sin \theta \) and \( -\cos \theta \). We need to integrate each part separately.

Step 2: Integrate \( \sin \theta \)

The integral of \( \sin \theta \) with respect to \( \theta \) is \( -\cos \theta \). This is a basic integration formula that you can find in an integration table.

Step 3: Integrate \( -\cos \theta \)

The integral of \( -\cos \theta \) with respect to \( \theta \) is \( -\sin \theta \). Again, this is a direct application of a known integration formula.

Step 4: Combine the Results

Combine the integrals of each component. The result is:\[-\cos \theta - \sin \theta + C\]where \( C \) is the constant of integration added because we are finding an indefinite integral.

Step 5: Write the Final Solution

The indefinite integral, after integrating both components and combining them, is:\[ \int (\sin \theta - \cos \theta) \, d\theta = -\cos \theta - \sin \theta + C \]

Key Concepts

Integral of Sine

The integral of a sine function, such as \( \int \sin \theta \, d\theta \), is one of the fundamental integrals you will encounter in calculus. When integrating \( \sin \theta \), the result is \( -\cos \theta \). This is because differentiation of \( -\cos \theta \) would bring us back to \( \sin \theta \), completing a circle of calculus operations.

Key points to remember:

• The integral of \( \sin \theta \) always results in \( -\cos \theta \), given the periodic nature of trigonometric functions.
• This result is derived from the fact that the derivative of \( \cos \theta \) is \( -\sin \theta \).

Understanding this basic integral is crucial as it serves as a foundation for solving more complex integration problems involving trigonometric functions.

Integral of Cosine

The integration of the cosine function is as straightforward as that of sine. The integral \( \int \cos \theta \, d\theta \) leads to \( \sin \theta \). This is because when differentiating \( \sin \theta \), the outcome is \( \cos \theta \). So, going in the reverse direction (i.e., integration), pulls us back to \( \sin \theta \).

Important points to keep in mind:

• The integral of \( \cos \theta \) results in \( \sin \theta \).
• This relationship is often helpful for various trigonometric integrations and useful in parsing more complex expressions.

Mastering the integral of cosine is essential for delving deeper into calculus topics and handling numerous mathematical problems efficiently.

Constant of Integration

In indefinite integrals, such as this one, the constant of integration, denoted usually by \( C \), plays a critical role. Because integration is essentially the reverse of differentiation, when you integrate a function, you need to account for any constant that might have been lost when the derivative was originally taken.

Here’s why it's important:

• It reflects an infinite number of possible antiderivatives that satisfy the derivative \( f'(x) \).
• For example, if two functions differ by a constant, their derivatives are the same. Thus, when you integrate, you add \( C \) to account for this.
• It ensures that your solution represents the entire family of antiderivatives, which could include an infinite number of possible curves vertically shifted along the y-axis.

Understanding the "constant of integration" is crucial in correctly writing solutions for indefinite integrals and capturing the complete set of solutions to the integral.