Question

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

Short answer

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

Step-by-step solution

Identify the Integral Form

Recognize that the integral \( \int \frac{1}{2x} dx \) can be rewritten as \( \frac{1}{2} \int \frac{1}{x} dx \). This is because the constant \( \frac{1}{2} \) can be factored out of the integral.

Integrate \( \frac{1}{x} \)

Recall the basic integral \( \int \frac{1}{x} dx = \ln |x| + C \), where \( C \) is the constant of integration. Applying this to our integral, it becomes \( \frac{1}{2} \ln |x| + C \).

Combine Results

Combine the results from the previous steps to complete the integral. The indefinite integral of \( \frac{1}{2x} \, dx \) is \( \frac{1}{2} \ln |x| + C \).

Key concepts

Integration Techniques

When dealing with indefinite integrals, recognizing the form of the integral can greatly simplify the calculation. In this example, we started with the integral \( \int \frac{1}{2x} \, dx \). The first step in solving such an integral involves simplifying it, if possible. Here, \( \frac{1}{2} \) is a constant factor that can be taken out of the integral. This operation transforms the integral into a more straightforward version: \( \frac{1}{2} \int \frac{1}{x} \, dx \).
Common integration techniques include:

• Substitution: Used when an integral can be transformed into a simpler form by changing the variable
• Integration by Parts: Useful for integrating products of functions
• Partial Fractions: Applied to rational expressions to break them down into simpler fractions

Understanding these techniques and their appropriate applications can make solving complex integrals more manageable. For the integral \( \int \frac{1}{x} \, dx \), a more basic technique involving recognizing a standard form was employed.

Logarithmic Function

The logarithmic function emerges naturally when integrating the reciprocal function, \( \frac{1}{x} \). In calculus, the integral \( \int \frac{1}{x} \, dx \) results in the natural logarithm, \( \ln |x| \). The absolute value signs are crucial because the logarithm of a negative number isn't defined in the real number system.
In the given problem, we tackled \( \int \frac{1}{x} \, dx \) by remembering this standard result. By integrating correctly, the natural logarithm function helps convert the variable \( x \) in a fraction into a more accessible form: \( \ln |x| \). Whenever you encounter similar integrals, recalling this relationship can speed up your calculations significantly. Additionally, it's essential to associate the logarithmic function with its property of transforming multiplicative relationships into additive ones, which is a central feature in calculus.

Constant of Integration

The constant of integration, denoted as \( C \), appears in every indefinite integral. It reflects the fact that there are infinitely many antiderivatives for a given function. Each value of \( C \) results in a different function, but they all share the same derivative.
When computing \( \int \frac{1}{2x} \, dx \), after factoring and integrating, we obtained \( \frac{1}{2} \ln |x| + C \). This \( C \) ensures that every possible antiderivative is accounted for. Without the constant of integration, the function would only represent one particular solution, ignoring the infinite others. In practical application, the constant becomes significant when solving initial value problems, or when additional conditions, like points the function must pass through, are given. It's always good practice to include \( C \) in your final answer unless specific initial conditions are provided.