Problem 27 - Calculus Early Transcendentals: Pearson New International Edition

Question

Perform the indicated integrations. $$ \int \frac{\sin x-\cos x}{\sin x} d x $$

Step-by-Step Solution

Answer

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

Step 1: Simplify the Integrand

The integrand \( \frac{\sin x - \cos x}{\sin x} \) can be simplified by splitting it into two separate fractions: \( \frac{\sin x}{\sin x} - \frac{\cos x}{\sin x} \). This simplifies further to \( 1 - \cot x \).

Step 2: Set Up the Integral with Simplified Integrand

Rewrite the integral using the simplified integrand from Step 1: \[\int (1 - \cot x) \, dx.\]

Step 3: Integrate the Separate Terms

Now integrate each term separately: \( \int 1 \, dx = x \), and for \( \int \cot x \, dx \), recall that \( \cot x = \frac{\cos x}{\sin x} \). This integral can be solved by recognizing \( \int \cot x \, dx = \ln|\sin x| \).

Step 4: Combine Results

Combine the results from each term. The integral becomes: \[x - \ln|\sin x| + C\] where \( C \) is the constant of integration.

Key Concepts

Trigonometric Integrals

Trigonometric integrals are integrals that involve trigonometric functions. In this exercise, we encountered the integral of an expression involving sine and cosine functions, \( \int \/frac{\sin x - \cos x}{\sin x} d x \)\. Understanding how to work with such expressions is vital in calculus.To handle trigonometric integrals, it's helpful to memorize the standard integrals of simple trigonometric functions:

• \( \int \sin x \, dx = -\cos x + C \)
• \( \int \cos x \, dx = \sin x + C \)
• \( \int \tan x \, dx = -\ln|\cos x| + C \)
• \( \int \cot x \, dx = \ln|\sin x| + C \)

Recognizing forms such as these allows one to simplify complex trigonometric integrals into manageable calculations. This makes the solution process more straightforward once you decompose the fractions as seen in the original solution.

Simplifying Integrands

Simplifying the integrand is a key step in making the integration process easier. In the given exercise, the integrand \( \frac{\sin x - \cos x}{\sin x} \) was simplified by splitting it into two separate fractions. This gives us \( \frac{\sin x}{\sin x} - \frac{\cos x}{\sin x} \), which further simplifies to \( 1 - \cot x \).Simplifying an integrand can reveal hidden patterns and make the integral of complex expressions more transparent. For instance, transforming fractions or combining like terms might unveil trigonometric identities that simplify the integration.In general, when tackling an integration problem, it's beneficial to:

• Identify and apply appropriate trigonometric identities.
• Separate terms as much as possible to simplify individually.
• Re-write the integrand in a form that is easier to integrate, as done with \( 1 - \cot x \) here.

This approach not only reduces the complexity of the task but also increases the likelihood of spotting easier integrals that can then be combined to form the solution.

Definite and Indefinite Integrals

When working with integrals in calculus, distinguishing between definite and indefinite integrals is important. The integral solved in the exercise is an indefinite integral, which results in a function that includes an arbitrary constant, represented as \( C \).

• **Indefinite Integrals:** These do not have specified limits of integration. They provide a family of functions, as they include a constant term \( C \). For example, \( \int (1 - \cot x) \, dx = x - \ln|\sin x| + C \).
• **Definite Integrals:** These have specific upper and lower limits on the integrals. The result is a numerical value, representing the area under the curve of the function from one point to another. No constant \( C \) is added, as the limits provide a fixed calculation.

Integrating with indefinite integrals as seen in this exercise is critical when determining general solutions or understanding function behavior. Conversely, definite integrals are employed in more practical scenarios, such as calculating areas or total accumulated quantities.