Problem 325 - Calculus Volume 1

Question

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

Step-by-Step Solution

Answer

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

Step 1: Recognize the Integral Form

The given integral is \( \int \frac{2}{x} \, dx \). This is a standard integral form where the integrand \( \frac{1}{x} \) is one of the basic integrals you should know. The integral of \( \frac{1}{x} \) with respect to \( x \) is \( \ln |x| \).

Step 2: Apply the Constant Multiple Rule

In the integral \( \int \frac{2}{x} \, dx \), the constant 2 can be factored out of the integral. This results in \( 2 \int \frac{1}{x} \, dx \).

Step 3: Integrate the Basic Form

Now integrate \( \int \frac{1}{x} \, dx \) which is known to be \( \ln |x| + C \), where \( C \) is the constant of integration.

Step 4: Multiply the Result by the Constant

Apply the constant multiple back to the result: \( 2 (\ln |x| + C) = 2\ln |x| + C' \). Here, \( C' \) still represents a constant of integration.

Key Concepts

Integral Calculus

Integral calculus is a fundamental branch of mathematics, focussing on the concept of integration which is essentially the reverse process of differentiation. It is used to find quantities like areas under curves, accumulated quantities, and more. In general, integration combines the values of a function over a range of values.
Understanding integrals is crucial because it allows us to solve problems related to calculating areas and volumes, among other applications in real-world scenarios.
The given problem asks us to compute an indefinite integral. Indefinite integrals describe a family of functions and include a constant of integration denoted by "\(C\)". This constant encompasses all possible vertical shifts of the antiderivative function, reflecting the many functions that can be derived from a given derivative.

Basic Integrals

Basic integrals are fundamental integrals that often serve as building blocks for solving more complex problems in calculus. One of the most essential basic integrals to remember is:

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

This integral is crucial in calculating the integral of any rational function that simplifies to a form resembling \(\frac{1}{x}\).
In our exercise, we see a classic example of using this basic integration rule: The integral \(\int \frac{2}{x} \, dx\) can be approached by recognizing it as a scaled version of \(\int \frac{1}{x} \, dx\). This step forms the cornerstone of solving such exercises by recalling these elementary integration formulas and applying them accordingly.

Constant Multiple Rule

The constant multiple rule is a simple yet powerful tool that helps to simplify integrals where multiplication by a constant is involved. If you have an integral of the form \(\int k \, f(x) \, dx\), where \(k\) is a constant, you can take \(k\) outside of the integral, simplifying it to \(k \int f(x) \, dx\).

• For example: \(\int 2 \times \frac{1}{x} \, dx\) simplifies to \(2 \int \frac{1}{x} \, dx\)

This adjustment makes it easier to apply known integral formulas directly to \(f(x)\), making calculations straightforward.
This concept is applied in our exercise when factoring out the constant "2" from \(\int \frac{2}{x} \, dx\), allowing us to directly integrate the remaining \(\frac{1}{x}\) term.

Natural Logarithm

The natural logarithm, denoted by \(\ln(x)\), is an important function in mathematics, defined as the logarithm to the base \(e\), where \(e\) is approximately 2.71828. The natural logarithm has unique properties, particularly its derivative which is exactly \(\frac{1}{x}\).
This intrinsic connection is evident in the integral \(\int \frac{1}{x} \, dx = \ln |x| + C\).
The absolute value in \(\ln |x|\) ensures the integral is defined for both positive and negative values of \(x\), excluding zero.
In the solution provided, recognizing the outcome of \(\int \frac{1}{x} \, dx\) as \(\ln |x|\) is key to determining the indefinite integral result for \(\int \frac{2}{x} \, dx\). The final result, \(2 \ln |x| + C\), showcases the combined application of both the constant multiple rule and the properties of the natural logarithm.