Question

In the following exercises, find each indefinite integral by using appropriate substitutions. $$ \int \frac{\cos x-x \sin x}{x \cos x} d x $$

Short answer

The indefinite integral is \( \ln \left( \frac{|x|}{|\cos x|} \right) + C \).

Step-by-step solution

Simplify the Integrand

Begin by simplifying the integrand \( \frac{\cos x - x \sin x}{x \cos x} \). This can be split into two separate fractions: \( \frac{\cos x}{x \cos x} - \frac{x \sin x}{x \cos x} \). Simplify each fraction to obtain \( \frac{1}{x} - \frac{\sin x}{\cos x} \).

Express the Integral

Rewrite the integral using the simplified form: \( \int \left( \frac{1}{x} - \frac{\sin x}{\cos x} \right) dx \). This simplifies to \( \int \frac{1}{x} \, dx - \int \tan x \, dx \).

Evaluate the First Integral

Evaluate the integral \( \int \frac{1}{x} \, dx \), which is \( \ln |x| + C_1 \) where \( C_1 \) is a constant of integration.

Evaluate the Second Integral

Evaluate the integral \( \int \tan x \, dx \). This requires using the identity \( \int \tan x \, dx = -\ln |\cos x| + C_2 \) where \( C_2 \) is a constant of integration.

Combine the Results

Combine results from Steps 3 and 4: \( \ln |x| - \ln |\cos x| + C \). Simplify further to express using a single logarithm: \( \ln \left( \frac{|x|}{|\cos x|} \right) + C \), where \( C \) is the constant of integration, \( C = C_1 - C_2 \).

Key concepts

Integration by Substitution

Integration by substitution is a method used to find indefine integrals, making the integration process easier by transforming the integrand into a simpler form. The purpose of substitution is to rewrite an integrand in terms of a single variable, rather than dealing with a composite function directly. Here's a step-by-step approach to understanding this technique:

• Choose the Substitution: Identify part of the integrand that complicates integration, often a composite function. Let this part be equal to a new variable, usually denoted by \( u \).
• Differentiate the Substitution: Find the derivative \( du \) of the substitution with respect to \( x \). This helps in transforming the entire integrand in terms of \( u \).
• Rewrite the Integral: Substitute both the integrand and the differentials in terms of \( u \). This step often simplifies the integral considerably.
• Solve the New Integral: With the integrand now in terms of \( u \), it often becomes a standard integral form.
• Resubstitute: Once integrated, replace \( u \) with the original variable to return to the original context of the problem.

In the given exercise, although substitution is not explicitly used in the step-by-step solution, recognizing trigonometric identities and simplifying using basic fraction rules reflect the fundamental idea behind substitution, which is to turn complex expressions into simpler forms suitable for integration.

Trigonometric Integrals

Trigonometric integrals often involve functions such as sine, cosine, tangent, and others, and require special techniques or identities to solve. These integrals take advantage of well-known trigonometric identities to simplify expressions.

In the exercise, the integral involves \( \tan x \), which is the ratio \( \frac{\sin x}{\cos x} \). After simplification of the integrand, the integral \( \int \tan x \, dx \) appears. To solve this, we use:

• Identity Usage: Recognize that integrating \( \tan x \) can be rephrased by using the identity \( \tan x = \frac{\sin x}{\cos x} \). This allows rearranging the problem to an easier form or using a known substitution.
• Standard Integral Forms: Understand the result of integrating \( \tan x \), which is \( -\ln |\cos x| \), helping to solve such integrals quickly through memorization of key integral results.

Successfully managing trigonometric integrals often requires practice with both identities and memorization of common integral results, allowing students to solve them with confidence.

Logarithmic Integration

Logarithmic integration deals with integrals that result in a natural logarithm function, typically arising when integrating functions of the form \( \frac{1}{x} \). These integrals are straightforward but crucial in understanding calculus at a deeper level.

In the context of the exercise, the integral \( \int \frac{1}{x} \, dx \) yields \( \ln |x| \). Here's how to handle logarithmic integration:

• Recognize the Form: Identify functions that can be expressed in the form \( \frac{1}{u} \) and link them to their integral being natural logarithm functions.
• Believe the Formula: Know that \( \int \frac{1}{x} \, dx = \ln |x| + C \), where \( C \) is a constant of integration. This is a well-established result crucial for solving many integrals.
• Combine Results: When integrating expressions involving logarithmic and other forms, combine them carefully. In our exercise, results are finally combined using the property of logarithms: \( \ln |x| - \ln |\cos x| = \ln \left( \frac{|x|}{|\cos x|} \right) \).

Logarithmic integration is often viewed as a stepping stone into understanding more complex calculus problems, providing essential insights into how integrals transform certain functions.