Problem 79 - Calculus for Biology and Medicine

Question

Compute the indefinite integrals. $$ \int \frac{1}{x+2} d x $$

Step-by-Step Solution

Answer

The integral is \( \ln |x+2| + C \).

Step 1: Identify the Integral Type

The given integral is \( \int \frac{1}{x+2} \, dx \). This is a basic form that can be solved using the natural logarithm integration rule.

Step 2: Apply the Natural Logarithm Rule

When we integrate \( \frac{1}{u} \), we get \( \ln |u| \). In our integral, \( u = x + 2 \). Therefore, the integral becomes \( \ln |x+2| + C \), where \( C \) is the constant of integration.

Key Concepts

Integration Techniques

Integration is the process of finding the antiderivative of a function. While there are many techniques available, choosing the right one is crucial for solving an integral efficiently. One common technique involves identifying the structure of the function. This allows for the application of specific integration rules. For the integral \( \int \frac{1}{x+2} \, dx \), recognizing it as a form that corresponds to the natural logarithm integration rule simplifies the process immensely.

Some other notable integration techniques include:

• Substitution: Used when an integral contains a composite function.
• Integration by parts: Applied when integrating a product of functions.
• Partial fraction decomposition: Useful for rational functions where the degree of the numerator is less than the degree of the denominator.
• Trigonometric integrals and substitution: Helpful with functions involving trigonometric identities.

Using these techniques effectively requires practice and a keen eye for recognizing the form of the integrand you are working with. Identifying and applying the correct technique streamlines the integration process.

Natural Logarithm Integration

The natural logarithm integration rule is a powerful tool for solving integrals of the form \( \int \frac{1}{u} \, du \). This specific structure simplifies to the natural logarithm: \( \ln |u| + C \), where \( C \) is the constant of integration. In our original exercise with \( \int \frac{1}{x+2} \, dx \), the function inside the integral is \( u = x + 2 \).

• The derivative of \( ln|u| \) is \( \frac{1}{u} \), justifying why the integral follows this pattern.
• The absolute value sign \( |u| \) is crucial because it ensures the logarithm is defined even for negative values.
• The rule applies to integrations where \( u \) is a linear function of \( x \), making it highly versatile in many basic computations.

Familiarity with this integration method can save time and simplify the solution process for a broad range of problems involving rational functions.

Constant of Integration

In indefinite integrals, we always include a constant of integration, commonly denoted as \( C \). This constant accounts for the family of antiderivatives that a function can have. When taking an antiderivative, any constant added will vanish upon differentiation. Thus, without \( C \), we cannot represent all possible solutions.

Some points to remember about \( C \):

• Every indefinite integral has an infinite number of antiderivatives, each differing by a constant \( C \).
• In definite integrals, \( C \) cancels out when evaluating the limits of integration, which is why it doesn't appear in the final answer.
• In real-world applications, determining \( C \) often requires additional initial condition information.

Understanding the presence and role of \( C \) is essential for solving indefinite integrals accurately. Neglecting \( C \) can lead to incomplete solutions and misunderstandings in both pure mathematics and applied contexts.