Problem 51 - Calculus for Biology and Medicine

Question

Use either substitution or integration by parts to evaluate each integral. $$ \int \frac{1}{\tan x} d x $$

Step-by-Step Solution

Answer

The integral \( \int \frac{1}{\tan x} \, dx \) is \( \ln |\sin x| + C \).

Step 1: Rewrite the Integral

Start by rewriting the integral \( \int \frac{1}{\tan x} \, dx \) using the identity \( \tan x = \frac{\sin x}{\cos x} \). This gives \( \int \cot x \, dx \) since \( \cot x = \frac{1}{\tan x} = \frac{\cos x}{\sin x} \).

Step 2: Use Substitution Method

Let \( u = \sin x \), then \( du = \cos x \, dx \). Substituting these into the integral, we get \( \int \frac{\cos x}{\sin x} \, dx \equiv \int \frac{1}{u} \, du \).

Step 3: Integrate \( \int \frac{1}{u} \, du \)

The integral of \( \frac{1}{u} \, du \) is \( \ln |u| + C \), where \( C \) is the constant of integration. Therefore, \( \int \frac{1}{u} \, du = \ln |u| + C \).

Step 4: Substitute Back to \( x \)

Since \( u = \sin x \), substitute back to get \( \ln |\sin x| + C \). Therefore, \( \int \cot x \, dx = \ln |\sin x| + C \).

Key Concepts

Substitution Method

The substitution method is a powerful technique used in calculus to simplify the process of integration. It involves replacing part of an integral with a new variable, making it easier to compute. In essence, it transforms a complex integral into a simpler one.
When using the substitution method, follow these general steps:

• Identify a substitution by selecting a part of the function inside the integral that can be replaced by a new variable, usually denoted as \( u \).
• Express \( du \), the derivative of \( u \), in terms of \( dx \).
• Modify the integral so that all parts are in terms of \( u \) and \( du \), eliminating \( x \) from the equation.
• Perform the integration with respect to \( u \).
• Substitute back to the original variable \( x \) after finding the indefinite integral.

In the original exercise, \( u = \sin x \) was chosen as the suitable substitution because its derivative, \( \cos x \, dx \), conveniently appeared in the integral expression. By using the substitution method, the integral simplifies to \( \int \frac{1}{u} \, du \), which is straightforward to integrate.

Trigonometric Integrals

Trigonometric integrals involve functions that include trigonometric expressions such as sine, cosine, or tangent. These integrals commonly require specific techniques to solve them, making them a unique part of calculus.
Trigonometric identities play a vital role here, as they can simplify the functions and make integration possible. For instance, recognizing that \( \cot x = \frac{\cos x}{\sin x} \) allowed for rewriting the integral in a manner that facilitated substitution.
Some common strategies for solving trigonometric integrals include:

• Using trigonometric identities to simplify expressions.
• Applying the substitution method to convert trigonometric terms into functions of a single variable.
• Utilizing formulas such as the integral of \( \cos \) and \( \sin \) functions, which are fundamental in calculus.

Mastering trigonometric integrals often requires practice to become familiar with the relationships and identities of trigonometric functions. In the exercise, converting \( \tan x \) to \( \cot x \) leveraged the identity, leading to a straightforward integration process using substitution.

Integration Techniques

Integration techniques encompass various methods for solving integrals, each applicable to different types of functions. Understanding these techniques is crucial in solving calculus problems efficiently.
Some integral techniques include:

• Substitution, which simplifies integrals using another variable.
• Integration by parts, which tackles products of functions and utilizes the formula: \[\int u \, dv = uv - \int v \, du\]
• Trigonometric identities and trigonometric substitution for handling complex trigonometric integrals.

In our specific exercise, the substitution method was suitable and effective, offering a clear path from the problem to the solution. Each kind of technique has its appropriate scenario, and choosing the right one can significantly ease the calculation process.
Remember, it's essential to become familiar with multiple techniques. This allows more flexibility in solving a wide variety of calculus integrals you might encounter in your studies.