Problem 76 - Thomas Calculus

Question

Derive the formula \(\sinh ^{-1} x=\ln \left(x+\sqrt{x^{2}+1}\right),-\infty

Step-by-Step Solution

Answer

The plus sign ensures positivity, necessary for real logarithmic values, deriving \( \sinh^{-1}(x) = \ln(x + \sqrt{x^2+1}) \).

Step 1: Understanding the Inverse Hyperbolic Sine

First, we need to understand what \( \sinh^{-1}(x) \) means. It represents the inverse function of hyperbolic sine function \( \sinh(y) = x \). Hence, \( y = \sinh^{-1}(x) \) implies \( x = \sinh(y) \), where \( \sinh(y) = \frac{e^y - e^{-y}}{2} \).

Step 2: Express \( x \) in terms of exponential functions

Starting from \( x = \sinh(y) \), set up the equation: \( x = \frac{e^y - e^{-y}}{2} \). Multiply through by 2 to clear the fraction, resulting in \( 2x = e^y - e^{-y} \).

Step 3: Rearrange and Square both sides

Rearrange terms to form: \( e^y - 2x = e^{-y} \). Square both sides of this equation to eliminate the negative exponent, leading to \((e^y - 2x)^2 = (e^{-y})^2 \). This simplifies to \( e^{2y} - 4xe^y + 4x^2 = e^{-2y} \).

Step 4: Solve for \( e^y \)

Note that \( e^{-2y} = \frac{1}{e^{2y}} \). Multiply your rearranged equation by \( e^{2y} \): \( e^{4y} - 4xe^{3y} + 4x^2e^{2y} = 1 \). Viewing this as a quadratic equation in terms of \( e^y \), solve to find \( e^y = x + \sqrt{x^2 + 1} \).

Step 5: Justification for plus/minus sign

Since the exponential function \( e^y \) must be positive, we choose the positive root \( e^y = x + \sqrt{x^2 + 1} \) to ensure positiveness, as the negative root does not satisfy this condition. Avoiding a negative root is necessary because it would lead to undefined or non-real solutions in the context of logarithms.

Step 6: Derive the inverse hyperbolic sine formula

Take the natural logarithm of both sides to isolate \( y \), resulting in \( y = \ln(e^y) = \ln(x + \sqrt{x^2+1}) \). Thus, \( \sinh^{-1}(x) = \ln(x + \sqrt{x^2+1}) \).

Key Concepts

Hyperbolic Functions

Hyperbolic functions are analogs of trigonometric functions that are used in many areas of mathematics. These functions include the hyperbolic sine, cosine, and others. They can be defined using exponential functions which makes them particularly useful in calculus and complex analysis.
The hyperbolic sine function, denoted by \( \sinh(y) \), is defined as:

• \( \sinh(y) = \frac{e^y - e^{-y}}{2} \)

This function is similar to the sine function from trigonometry, but it uses exponential terms rather than angles on a unit circle. It's widely used in solving problems involving hyperbolas and in engineering contexts.
Inverse hyperbolic sine, \( \sinh^{-1}(x) \), is the inverse of this function. This means, if \( y = \sinh^{-1}(x) \), then \( x = \sinh(y) \). Understanding this relationship is crucial for converting between algebraic forms and interpretations that involve these types of functions.

Exponential Functions

Exponential functions are mathematical functions of the form \( f(y) = e^y \), where \( e \) is Euler’s number, approximately 2.718. These functions are crucial in modeling growth processes, such as population growth and compound interest. Their distinctive feature is the constant multiplicative rate of change.
In the context of hyperbolic functions, exponentials are used to define and solve these equations. Specifically, in the hyperbolic sine function \( \sinh(y) = \frac{e^y - e^{-y}}{2} \), the presence of \( e^y \) and \( e^{-y} \) represents the behavior of hyperbolas.
To solve for \( e^y \) from \( \sinh(y) = x \), we rearrange the equation \( x = \frac{e^y - e^{-y}}{2} \) to extract \( e^y \) in terms of \( x \). This results in \( e^y = x + \sqrt{x^2 + 1} \), highlighting the necessity of choosing the positive root to maintain the function's positive nature.

Logarithmic Functions

Logarithmic functions, such as \( \ln(y) \), are the inverses of exponential functions. When you have an equation involving an exponential function, you often solve for the exponent using logarithms. The natural logarithm \( \ln \) specifically uses the base \( e \).
In our case, after deriving \( e^y = x + \sqrt{x^2 + 1} \), taking the natural logarithm helps isolate \( y \). This is because \( y = \ln(e^y) \) simplifies to \( y = \ln(x + \sqrt{x^2 + 1}) \).
Thus, the inverse hyperbolic sine \( \sinh^{-1}(x) \) can be represented in a logarithmic form:

• \( \sinh^{-1}(x) = \ln(x + \sqrt{x^2 + 1}) \)

Understanding logarithmic functions is essential as they provide a powerful tool for solving equations where the unknown variable is an exponent. They help to transform complex multiplicative relationships into simpler additive ones, which are easier to solve and analyze.