Problem 27 - Calculus Early Transcendentals: Pearson New International Edition

Question

In Problems 1-54, perform the indicated integrations. \(\int \frac{\sin x-\cos x}{\sin x} d x\)

Step-by-Step Solution

Answer

The integrated result is \( x - \ln|\sin x| + C \).

Step 1: Simplify the Integrand

The given integral is \( \int \frac{\sin x - \cos x}{\sin x} dx \). We start by simplifying the integrand by splitting it into two separate fractions: \( \frac{\sin x}{\sin x} - \frac{\cos x}{\sin x} \). This simplifies further to \( 1 - \cot x \). Thus, the integral becomes \( \int (1 - \cot x) dx \).

Step 2: Integrate Each Term Separately

Now, we integrate each term separately. The integral of \( 1 \) with respect to \( x \) is simply \( x \), since the derivative of \( x \) is \( 1 \). Next, the integral of \(- \cot x \) with respect to \( x \) is \( \ln|\sin x| \), because the derivative of \( \ln|\sin x| \) is \(-\cot x \).

Step 3: Combine and Simplify the Result

Combine the results of the separate integrals to write the final integrated expression: \( x - \ln|\sin x| + C \), where \( C \) is the constant of integration. Make sure not to forget this constant, as it is an essential part of indefinite integrals.

Key Concepts

Trigonometric Integration

Trigonometric integration is a method used to integrate functions involving trigonometric expressions such as sine, cosine, and tangent. This technique is crucial in calculus because trigonometric functions often appear in real-world applications. Understanding how to work with these functions is essential for solving a wide variety of problems.

When faced with an integral involving trigonometric functions, there are several key approaches you can take:

• Look for opportunities to use trigonometric identities, such as the Pythagorean identity, to simplify expressions.
• Consider rewriting the integrand in terms of a single trigonometric function.
• Be aware of inverse trigonometric functions that might simplify the integration process.

For instance, in the integral \( \int \frac{\sin x - \cos x}{\sin x} dx \), identifying that \( \sin x \) and \( \cos x \) can be simplified independently using identities can expedite the integration process away from more complex patterns.

Integrand Simplification

Integrand simplification involves making the expression we intend to integrate less complicated and easier to handle. This can often be achieved by breaking it down into smaller and more manageable pieces. This step is crucial because it often allows us to see clearly through the problem.

The simplification process might involve:

• Dividing complex expressions into separate terms, as demonstrated in the original problem where \( \frac{\sin x - \cos x}{\sin x} \) becomes \( 1 - \cot x \).
• Using algebraic techniques to factor out common terms or cancel terms directly.
• Leveraging trigonometric identities to transform expressions into standard integrable functions.

By simplifying \( \frac{\sin x - \cos x}{\sin x} \) into \( 1 - \cot x \), the problem becomes straightforward, allowing easy application of basic integration techniques, avoiding unnecessary complications.

Integration Techniques

Integration techniques are fundamental tools that help in evaluating integrals, especially when straightforward anti-differentiation is not possible. They provide a variety of methods to handle different forms of integrals, particularly when they involve trigonometric functions.

In general, when integrating functions like \( \int (1 - \cot x) dx \), you can apply:

• Basic Integration Rules: Techniques such as linearity can be applied, where you integrate each term separately allowing you to manage each part on its own.
• Substitution: Often, substitution simplifies the integral significantly, especially with trigonometric expressions where direct integration is cumbersome.
• Transcendental Functions: Recognizing when a function is part of a transcendental, like \( \ln|\sin x| \), helps in quickly determining the integral due to a known derivative.

In our example, first integrating \( 1 \) yields \( x \), and integrating \(-\cot x \) yields \( \ln|\sin x| \), making \( \int (1 - \cot x) dx = x - \ln|\sin x| + C \). Always remember to include the constant of integration \(C\), as it accounts for any constant value that could be added to the indefinite integral.