91Ó°ÊÓ

The natural logarithm, denoted as \( \ln \), is a fundamental concept in calculus and mathematics in general. It is often associated with the constant \( e \), which is approximately 2.71828. When we take the natural logarithm of a number, we are essentially asking "what power must \( e \) be raised to, to obtain this number?"

Natural logarithms are used extensively in various mathematical expressions and are vital in solving equations involving exponential growth or decay. For example, \( \ln(1+x) \) can be expressed as an integral:

• \( \ln(1+x) = \int_0^x \frac{1}{1+t} \, dt \)

This integral form can be useful in deriving series expansions and is crucial when working with functions like \( \ln \left( \frac{1+x}{1-x} \right) \).

Partial fraction decomposition is a technique that helps simplify complex rational expressions. It's particularly useful when you need to integrate functions that contain polynomial ratios. The goal here is to express a complicated fraction as a sum of simpler fractions.

For example, consider the expression \( \frac{2}{1-t^2} \), which can be decomposed using:

• \( \frac{2}{1-t^2} = \frac{1}{1-t} + \frac{1}{1+t} \)

This decomposition is especially useful when evaluating integrals or series expansions because it allows each term to be handled separately.

In our problem, using partial fraction decomposition simplifies the integral of \( \frac{2}{1-t^2} \) to a form suitable for direct integration, which is key to transforming the integral into a recognizable function or series.

Series expansion is a powerful tool in mathematics used to approximate complex functions through a sum of simpler terms. A common type of series is the Maclaurin series, which is a Taylor series expansion about zero.

The Maclaurin series for a function \( f(x) \) can be expressed as:

• \( f(x) = f(0) + f'(0)x + \frac{f''(0)x^2}{2!} + \frac{f'''(0)x^3}{3!} + \cdots \)

For our function \( \text{atanh}(x) \), the series expansion is

• \( \text{atanh}(x) = x + \frac{x^3}{3} + \frac{x^5}{5} + \cdots \)

This infinite series effectively approximates \( \text{atanh}(x) \) over the interval \(|x| < 1\).
This concept is fundamental for mathematicians and engineers who often deal with functions that are too complex to work with directly but are manageable when expressed as a series.

Inverse hyperbolic functions, like \( \text{atanh}(x) \), are analogous to inverse trigonometric functions but appear in hyperbolic contexts. They are important in different areas of mathematics, including calculus and complex analysis.

The function \( \text{atanh}(x) \), or inverse hyperbolic tangent, is defined as:

• \( \text{atanh}(x) = \frac{1}{2} \ln \left( \frac{1+x}{1-x} \right) \)

In the given exercise, you saw it utilized in the expression \( \ln \left( \frac{1+x}{1-x} \right) \). Understanding how to manipulate such functions with integrals and series expansions is crucial
because they frequently show up in practical applications, such as in physics, engineering, and even in statistical models.
These functions help simplify complex expressions and provide a link between algebraic formulas and their geometrical or physical interpretations.