Problem 76 - Thomas Calculus in SI Units

Question

Derive the formula \(\sinh ^{-1} x=\ln (x+\sqrt{x^{2}+1})\) for all real \(x\) Explain in your derivation why the plus sign is used with the square root instead of the minus sign.

Step-by-Step Solution

Answer

The plus sign ensures positivity, as \(e^y > 0\).

Step 1: Understand the Hyperbolic Sine Inverse

The hyperbolic sine inverse function is defined as \( anh^{-1} x = y\) which means that \(x = \sinh y\). Therefore, we can write \(x = \frac{e^y - e^{-y}}{2}\). Our goal is to express \(y\) in terms of \(x\).

Step 2: Rearrange the Hyperbolic Sine Equation

Multiply both sides of the equation \(x = \frac{e^y - e^{-y}}{2}\) by 2 to eliminate the fraction, resulting in \(2x = e^y - e^{-y}\).

Step 3: Substitute \(e^{-y}\) with \(1/e^y\)

Rearrange to get \(2x = e^y - \frac{1}{e^y}\). Then, multiply the entire equation by \(e^y\) to remove the fraction: \(2x e^y = (e^y)^2 - 1\).

Step 4: Form a Quadratic Equation

Substitute \(z = e^y\), then the equation becomes \(2xz = z^2 - 1\) or \(z^2 - 2xz - 1 = 0\), which is a quadratic equation in terms of \(z\).

Step 5: Solve the Quadratic Equation

Use the quadratic formula to solve for \(z\): \[ z = \frac{2x \pm \sqrt{(2x)^2 + 4}}{2} = x \pm \sqrt{x^2 + 1} \]

Step 6: Select the Correct Sign

Given \(z = e^y\), it must be positive because \(e^y > 0\) for all real \(y\). Therefore, we choose the positive sign: \[ z = x + \sqrt{x^2 + 1} \]

Step 7: Find the Natural Logarithm

Since \(z = e^y\), take the natural logarithm of both sides to isolate \(y\): \[ y = \ln(z) = \ln(x + \sqrt{x^2 + 1}) \]

Step 8: Confirm the Result

Therefore, the inverse hyperbolic sine function is: \[ \sinh^{-1}(x) = \ln(x + \sqrt{x^2 + 1}) \]

Key Concepts

Understanding Hyperbolic Functions

Hyperbolic functions are analogs of the traditional trigonometric functions but for a hyperbola, much like trigonometric functions are for a circle. Among these functions, the hyperbolic sine function, denoted as \( \sinh(x) \), plays a vital role. Its formula is given by \( \sinh(x) = \frac{e^x - e^{-x}}{2} \), where \( e \) is the base of the natural logarithm.
Hyperbolic functions are essential in many areas of mathematics and physics, including calculus and complex analysis. They help describe certain curves, motions, and natural phenomena like heat distribution and waves.
In the context of the inverse hyperbolic sine, our objective is to express a real number \( x \) in terms of a hyperbolic angle \( y \), where \( x = \sinh(y) \). To solve for \( y \), the inverse function \( \sinh^{-1}(x) \) is used. This function provides a direct way to find \( y \) when \( x \) is known. The formula for \( \sinh^{-1}(x) \) is particularly interesting because it connects hyperbolic functions with logarithmic expressions, bridging parts of mathematics that might otherwise seem unrelated.

Deriving from a Quadratic Equation

To understand how the inverse hyperbolic sine can be expressed using a logarithmic function, we must first form and solve a quadratic equation.
Starting with the definition of the hyperbolic sine function and making the necessary algebraic manipulations, we arrive at an equation of the form \( z^2 - 2xz - 1 = 0 \), where \( z = e^y \). This equation is quadratic in terms of \( z \).
The quadratic formula, \( z = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \), is vital for solving this type of equation. Here, since our potential solutions for \( z \) are expressed in terms of \( x \), we get:

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

It is crucial to choose the correct sign. Since \( z = e^y \) represents an exponential function, it must be positive. Thus, we select the positive solution: \( z = x + \sqrt{x^2 + 1} \). This step ensures that our found solution remains valid for real numbers.

The Role of the Natural Logarithm

The natural logarithm, denoted as \( \ln \), is an indispensable function in mathematics, especially when dealing with exponentials and growth processes. In this derivation, the natural log helps unravel the expression for \( y \) in terms of \( x \) when we have \( e^y = z \).
By taking the natural logarithm of both sides of \( z = e^y \), we get \( y = \ln(z) \). Importantly, \( y \) is what we are solving for in the expression \( \sinh^{-1}(x) \).
When \( z \) is replaced by its correct positive form, \( x + \sqrt{x^2 + 1} \), we derive:

• \( y = \ln(x + \sqrt{x^2 + 1}) \)

This connection illustrates how exponential relationships and their inverses, through logarithms, allow us to solve seemingly complex functions like the inverse hyperbolic sine integral to advanced topics in mathematics. The natural logarithm thereby not only supports the exploration of hyperbolic functions but also simplifies the understanding of their inverses.