91Ó°ÊÓ

In mathematics, the natural logarithm is a logarithm to the base of the constant 'e', where 'e' is approximately equal to 2.71828. The natural logarithm is commonly denoted as ln(x). The natural logarithm of a number x is the power to which 'e' has to be raised to obtain the number x. For example, \(\text{ln}(e) = 1\) because \(e^1 = e\).

Natural logarithms (ln) are especially useful in calculus, as they have properties that simplify integration and differentiation. Here are some key properties:

• \( \text{ln}(a \times b) = \text{ln}(a) + \text{ln}(b) \)
• \( \text{ln}\bigg(\frac{a}{b}\bigg) = \text{ln}(a) - \text{ln}(b) \)
• \( \text{ln}(a^b) = b \times \text{ln}(a) \)

One very important application of the natural logarithm is in transforming certain integrals for easier evaluation. In our problem, we express the hyperbolic arctangent function in terms of a natural logarithm to simplify the integral calculation.

The hyperbolic arctangent function, denoted as arctanh(x), is the inverse function of the hyperbolic tangent function, tanh(x). The hyperbolic tangent function is defined as:

\[ \text{tanh}(x) = \frac{\text{sinh}(x)}{\text{cosh}(x)} = \frac{e^x - e^{-x}}{e^x + e^{-x}} \]

The arctanh(x) function is defined for values of x in the interval (-1, 1), and it is given by:

\[ \text{arctanh}(x) = \frac{1}{2} \text{ln}\bigg(\frac{1+x}{1-x}\bigg) \]

This expression is particularly useful because it allows us to express the result of an integral in terms of the natural logarithm. In our problem, recognizing that \( \frac{d}{dx}\text{arctanh}(x) = \frac{1}{1-x^2} \) helped us identify the integral and simplify it using the properties of arctanh.

Integrating functions involves finding the antiderivative, which is a function whose derivative gives back the original function. Several key properties and rules can simplify the integration process:

• Linearity: \[ \text{If } f(x) \text{ and } g(x) \text{ are integrable functions, then } \ \text{∫}[af(x) + bg(x)]dx = a\text{∫}f(x)dx + b\text{∫}g(x)dx \ \text{ for constants } a \text{ and } b. \]
• Integral of a sum: \[ \text{∫}[f(x) + g(x)]dx = \text{∫}f(x)dx + \text{∫}g(x)dx \]
• Substitution Method: Useful for transforming a tricky integral into a simpler form. Typically used when dealing with composite functions.
• Integration by Parts: Inverse of the product rule. If \[ u = f(x) \text{ and } dv = g'(x)dx \ then, \ \text{∫}udv = uv - \text{∫}vdu \]

In our particular problem, recognizing the integral of \(\frac{1}{1-x^2}\) as the derivative of arctanh(x) made the evaluation straightforward. Additionally, using the property \( \text{arctanh}(-x) = -\text{arctanh}(x) \) helped further simplify the definite integral calculation.