Problem 324 - Calculus Volume 1

Question

In the following exercises, compute each indefinite integral. $$\int \frac{1}{2 x} d x$$

Step-by-Step Solution

Answer

\( \frac{1}{2} \ln |x| + C \)

Step 1: Identify the Integral Form

The given integral is \( \int \frac{1}{2x} \, dx \). This integral can be identified as a basic form of a natural logarithm integral, because \( \frac{1}{2x} \) can be rewritten as \( \frac{1}{2} \cdot \frac{1}{x} \).

Step 2: Apply the Basic Integral Formula

Recognize that \( \int \frac{1}{x} \, dx = \ln |x| + C \), where \( C \) is the constant of integration. Therefore, the integral \( \int \frac{1}{2} \cdot \frac{1}{x} \, dx \) can be computed using this formula.

Step 3: Factor Out Constants

Take the constant \( \frac{1}{2} \) outside the integral: \( \int \frac{1}{2x} \, dx = \frac{1}{2} \int \frac{1}{x} \, dx \).

Step 4: Compute the Integral

Now calculate the integral: \( \frac{1}{2} \int \frac{1}{x} \, dx = \frac{1}{2} \ln |x| + C \).

Step 5: Write the Final Solution

Combine everything to write the final solution: \( \frac{1}{2} \ln |x| + C \), where \( C \) is a constant of integration.

Key Concepts

Natural Logarithm

The natural logarithm, denoted as \( \ln(x) \), is an essential concept in calculus, particularly when dealing with integrals that involve expressions like \( \frac{1}{x} \). It is the inverse operation to the exponential function \( e^x \) and has a variety of useful properties. In terms of integration, the integral \( \int \frac{1}{x} \, dx \) results in the natural logarithm \( \ln |x| \). This is because the derivative of \( \ln |x| \) is \( \frac{1}{x} \).

**Key Points about Natural Logarithms:**

• The argument \( x \) of \( \ln(x) \) must be positive in real-valued functions to ensure that the logarithm remains defined.
• When integrating expressions that simplify to \( \frac{1}{x} \), always remember the result involves a natural logarithm and requires an absolute value, \( \ln |x| \), to cater to all possible values of \( x \).
• The constant \( e \) (approximately 2.71828) is known as Euler's number and is the base of natural logarithms.

Natural logarithms are very useful across various fields of mathematics and science, from calculating growth and decay processes to analyzing complex functions.

Constant of Integration

In the context of indefinite integrals, the constant of integration, denoted \( C \), plays a crucial role. When we compute an indefinite integral, there are an infinite number of functions (differing by a constant) whose derivative is the integrand. Therefore, an indefinite integral can yield multiple solutions based on this arbitrary constant.

**Why the Constant of Integration is Important:**

• It reflects the idea that integrating a function results in a family of functions.
• The constant accounts for all possible vertical shifts of the antiderivative along the y-axis.
• In physical applications, \( C \) can represent initial conditions, such as starting amounts or initial velocities.

For example, when we integrate \( \frac{1}{x} \), the result is \( \ln |x| + C \). Without \( C \), we would be limiting the solution to a single function, ignoring the infinite possibilities that integration naturally implies.

Integral Calculus

Integral calculus is a fundamental branch of calculus that focuses on two key operations: determining areas under curves and finding antiderivatives (or integrals). Integral calculus is divided into indefinite integrals and definite integrals.

**Understanding Indefinite Integrals:**
Indefinite integrals do not have limits of integration. The result of an indefinite integral is a function (plus a constant of integration \( C \)) that represents all the antiderivatives of the original function.

**Applications of Integral Calculus:**

• Calculating areas and volumes is one of the most practical uses of integral calculus in physics and engineering.
• Solving differential equations often involves finding the integral of a derivative function.
• Integral calculus is used in probability to determine cumulative distribution functions.

Understanding integral calculus and its principles can reveal insights into both theoretical and practical problems, making it an irreplaceable tool in many scientific and engineering fields. It lays the groundwork for advanced analysis and problem-solving techniques beyond basic differentiation.