Unlock ln(x) Secrets: The Antiderivative Explained!

Integration by parts, a powerful technique in calculus, provides the key to unlocking the mysteries surrounding the antiderivative of ln(x). Specifically, exploring functions requires a solid understanding of mathematical software like Mathematica for verifying solutions and performing complex calculations. The concept of natural logarithm, denoted as ln(x), plays a fundamental role, and the rules defined by Gottfried Wilhelm Leibniz, a notable mathematician and logician, are essential when exploring its antiderivative.

Unveiling the Mystery: The Antiderivative of ln(x) Explained!

This article aims to comprehensively explain how to find the antiderivative of the natural logarithm function, ln(x). We will break down the process step-by-step, providing clear explanations and examples. The core focus will remain on understanding and applying the technique to solve for the antiderivative of ln(x).

Understanding the Basics: ln(x) and Antiderivatives

Before diving into the calculation, let’s ensure a solid foundation. We’ll review what ln(x) represents and what taking an antiderivative means.

What is ln(x)?

• ln(x) represents the natural logarithm of x.
• It answers the question: "To what power must the number e (Euler’s number, approximately 2.71828) be raised to equal x?".
• The domain of ln(x) is all positive real numbers (x > 0).
• Understanding the graph of ln(x) is helpful. It starts at negative infinity as x approaches 0, passes through the point (1,0), and increases slowly as x increases.

What is an Antiderivative?

• An antiderivative of a function f(x) is a function F(x) whose derivative is f(x). Mathematically, F'(x) = f(x).
• Finding an antiderivative is also known as integration.
• Antiderivatives are not unique. We add a constant of integration, usually denoted by "C", because the derivative of a constant is always zero. Therefore, if F(x) is an antiderivative of f(x), then F(x) + C is also an antiderivative.
• The antiderivative of f(x) is represented as ∫f(x) dx.

Finding the Antiderivative of ln(x): Integration by Parts

The standard method for finding the antiderivative of ln(x) is integration by parts.

The Integration by Parts Formula

Integration by parts is based on the product rule for differentiation. It states:

∫u dv = uv – ∫v du

Where:

• u and v are functions of x.
• du is the derivative of u.
• dv is the derivative of v.

The trick is choosing the appropriate u and dv.

Applying Integration by Parts to ln(x)

1. Choosing u and dv:

   • Let u = ln(x)
   • Let dv = dx

   The reasoning behind this choice is that the derivative of ln(x) is a simple expression (1/x), which will simplify the subsequent integration.

2. Finding du and v:

   • du = (1/x) dx (Derivative of ln(x))
   • v = x (Antiderivative of dx, i.e., ∫dx = x)

3. Applying the Formula:

   ∫ln(x) dx = (ln(x))(x) – ∫x (1/x) dx

4. Simplifying:

   ∫ln(x) dx = x ln(x) – ∫1 dx

5. Integrating ∫1 dx:

   ∫1 dx = x

6. The Final Result:

   ∫ln(x) dx = x ln(x) – x + C

   Therefore, the antiderivative of ln(x) is x ln(x) – x + C.

Verification: Differentiating the Result

To verify that we have correctly found the antiderivative, we can differentiate our result:

d/dx [x ln(x) – x + C]

Applying the product rule to x ln(x):

d/dx [x ln(x)] = (1)(ln(x)) + x(1/x) = ln(x) + 1

Differentiating the rest of the expression:

d/dx [-x + C] = -1 + 0 = -1

Combining these results:

ln(x) + 1 – 1 = ln(x)

Since the derivative of x ln(x) – x + C is ln(x), our antiderivative is correct.

Examples of Applying the Antiderivative of ln(x)

Here are a couple of examples demonstrating how to use the antiderivative of ln(x).

Example 1: Definite Integral

Evaluate the definite integral: ∫₁^e ln(x) dx

1. We already know that the antiderivative of ln(x) is x ln(x) – x + C. We can drop the constant C when working with definite integrals.

2. Apply the Fundamental Theorem of Calculus:

   ∫₁^e ln(x) dx = [e ln(e) – e] – [1 ln(1) – 1]

3. Simplify:

   • ln(e) = 1
   • ln(1) = 0

   Therefore:

   ∫₁^e ln(x) dx = [e(1) – e] – [1(0) – 1] = [e – e] – [0 – 1] = 0 – (-1) = 1

Example 2: Indefinite Integral with a Constant Multiple

Find the antiderivative of 3 ln(x).

1. Use the constant multiple rule: ∫kf(x) dx = k∫f(x) dx, where k is a constant.

   ∫3 ln(x) dx = 3 ∫ln(x) dx

2. We know ∫ln(x) dx = x ln(x) – x + C

3. Substitute:

   3 ∫ln(x) dx = 3 [x ln(x) – x + C]

4. Distribute and simplify (remembering that 3C is still a constant, which we can represent as C):

   3x ln(x) – 3x + 3C = 3x ln(x) – 3x + C

   Therefore, the antiderivative of 3 ln(x) is 3x ln(x) – 3x + C.

So there you have it! Mastering the antiderivative of ln(x) might seem tricky at first, but with a little practice, you’ll be solving these integrals like a pro. Now go forth and conquer those calculus problems!