Question

For the following exercises, find the definite or indefinite integral. $$ \int \frac{\cos x d x}{\sin x} $$

Short answer

\( \int \frac{\cos x}{\sin x} \, dx = \ln |\sin x| + C \)

Step-by-step solution

Identify the Type of Integral

The given integral is \( \int \frac{\cos x}{\sin x} \, dx \), which involves a trigonometric function. This suggests the use of trigonometric identities or substitution for simplification.

Use Trigonometric Identity

Recognize that \( \frac{\cos x}{\sin x} \) can be rewritten using the trigonometric identity as \( \cot x \), so the integral becomes \( \int \cot x \, dx \).

Recall the Integral of Cotangent

The integral \( \int \cot x \, dx = \ln |\sin x| + C \), where \( C \) is the constant of integration. This is a standard integral result.

Write the Final Answer

Therefore, the indefinite integral is given by \( \int \frac{\cos x}{\sin x} \, dx = \ln |\sin x| + C \).

Key concepts

Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that are true for every value of the variable where both sides of the equation are defined. These identities are incredibly useful when solving integrals because they allow us to simplify expressions. For instance, in the integral \( \int \frac{\cos x}{\sin x} \ dx \), you can notice right away that \( \frac{\cos x}{\sin x} \) is simply \( \cot x \). Recognizing this identity helps in transforming a complex looking integral into a much simpler one, which can be easily integrated.

• Common Trigonometric Identities:
• \( \sin^2 x + \cos^2 x = 1 \)
• \( 1 + \tan^2 x = \sec^2 x \)
• \( 1 + \cot^2 x = \csc^2 x \)

These identities are often the key to solving integrals involving trigonometric functions, making them an essential tool in calculus.

Substitution Method

The substitution method is a powerful technique in integration used to simplify integrals that are difficult to solve at first glance. By substituting a part of the integral with a new variable, the integral can often be transformed into a simpler form. This method is similar to the chain rule in differentiation but applied in reverse.Consider the integral \( \int \frac{\cos x}{\sin x} \ dx \). Once we recognized \( \frac{\cos x}{\sin x} \) as \( \cot x \), it becomes evident that its integral is a standard result. However, if this recognition wasn't immediate, substitution could help simplify it. For example:

• Substitute \( u = \sin x \), then \( du = \cos x \ dx \).
• The integral becomes \( \int \frac{1}{u} \, du \), which is a basic logarithm integral.

This shows the power of substitution, where a seemingly complex integral turns into an easy integration problem.

Integration Techniques

Integration techniques are methods used to solve integrals that do not naturally fit into simple forms. These techniques include direct integration, substitution, integration by parts, and sometimes special functions or numerical methods.The exercise with \( \int \frac{\cos x}{\sin x} \ dx \) showcases using direct integration after simplifying with trigonometric identities. Once simplified to \( \int \cot x \, dx \), the direct integration results in a known formula:

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

In solving integrals:

• First, look for any possible simplifications using trigonometric identities or algebraic manipulations.
• Consider substitution if simplification isn't straightforward.
• Recognize direct integral results or apply integration by parts when appropriate.

Mastering these techniques requires practice and familiarity with a variety of integrals, ensuring you can tackle problems from different angles for the best solutions.