Problem 79 - Calculus for Biology and Medicine

Question

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

Step-by-Step Solution

Answer

\( \int \frac{1}{x+2} \, dx = \ln |x+2| + C \)

Step 1: Recognize the Integral Form

Notice that the integral is in the form \( \int \frac{1}{x+a} \, dx \), which is a basic logarithmic integral.

Step 2: Apply the Logarithmic Integration Rule

Use the rule \( \int \frac{1}{x+a} \, dx = \ln |x+a| + C \), where \( a = 2 \).

Step 3: Substitute Back into the General Solution

Substitute the specific value \( a = 2 \) into the rule to get the solution \( \int \frac{1}{x+2} \, dx = \ln |x+2| + C \).

Key Concepts

Logarithmic Integration

Logarithmic integration is a method used to solve integrals that involve logarithmic functions, especially those that take the form \( \int \frac{1}{x+a} \, dx \). Here, the key is recognizing that the integral of \( \frac{1}{x+a} \) is related to natural logarithms, which are a fundamental part of calculus.
To solve these types of integrals, we apply the logarithmic integration rule, which is:

• The integral \( \int \frac{1}{x+a} \, dx = \ln |x+a| + C \), where \( C \) is the constant of integration.

The constant of integration represents the arbitrary constant that arises when finding antiderivatives, ensuring that all solutions to the indefinite integral are accounted for. By understanding this rule, you can efficiently solve problems involving logarithmic integration.
Logarithmic integrals are particularly helpful when functions cannot be simplified into a polynomial form, making them an essential aspect of calculus.

Basic Integrals

Basic integrals refer to fundamental integration problems that often have direct solutions. These solutions don't usually require complex transformation or substitution techniques. Understanding these can simplify many calculus problems.
For instance, a classic example is the integral \( \int x^n \, dx \). The basic rule applied here is:

• \( \int x^n \, dx = \frac{x^{n+1}}{n+1} + C \), provided \( n eq -1 \)

In the case of an integral with a denominator as seen in \( \int \frac{1}{x+a} \, dx \), the logarithmic rule we used is categorized as a basic integral because it can be computed directly once the pattern is recognized. This makes basic integrals an important step in solving more advanced calculus problems later.

Calculus Problems

Calculus problems involve finding derivatives or integrals of functions and applying these concepts to solve real-world problems. Integrals help in finding areas under curves, computing total accumulations, and understanding dynamic systems.
In calculus, problems often require recognition of different integral types for effective problem-solving. Recognizing forms like \( \int \frac{1}{x+a} \, dx \) enables you to employ the correct rule right away. For this specific case, you apply the logarithmic integration method.
Solving calculus problems requires practice, a good understanding of fundamentals like basic integrals, and the ability to identify patterns. With these skills, you can tackle a wide range of calculus problems with confidence. Additionally, grasping these concepts prepares you for applications in physics, engineering, and other sciences where calculus is extensively used.