Problem 76 - Thomas Calculus

Question

Derive the formula \(\sinh ^{-1} x=\ln \left(x+\sqrt{x^{2}+1}\right)\) 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 formula \( \sinh^{-1} x = \ln(x+\sqrt{x^2+1}) \) is derived, using the positive root for \( e^y \) to ensure a positive exponential.

Step 1: Define the Inverse Hyperbolic Function

Start with the definition of the inverse hyperbolic sine function: for a given real number \( y \), \( y = \sinh^{-1}(x) \) implies \( x = \sinh(y) \). This definition will help in expressing \( x \) in terms of \( y \).

Step 2: Express sinh(y) Using Its Definition

The hyperbolic sine function \( \sinh(y) \) is defined as \( \sinh(y) = \frac{e^y - e^{-y}}{2} \). Since we have \( x = \sinh(y) \), this can be written as \( x = \frac{e^y - e^{-y}}{2} \), which can be rearranged to \( e^y - e^{-y} = 2x \).

Step 3: Solve for e^y

Multiply both sides of the equation \( e^y - e^{-y} = 2x \) by \( e^y \) to eliminate the fraction: \( e^{2y} - 1 = 2xe^{y} \). This can be rewritten as a quadratic equation in terms of \( e^y \): \( e^{2y} - 2xe^y - 1 = 0 \).

Step 4: Apply the Quadratic Formula

Use the quadratic formula \( e^y = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \) to solve for \( e^y \), where \( a = 1 \), \( b = -2x \), and \( c = -1 \). This gives: \[e^y = \frac{2x \pm \sqrt{(2x)^2 - 4 \cdot 1 \cdot (-1)}}{2}\sin.\] Simplifying this, we obtain \( e^y = x \pm \sqrt{x^2 + 1} \).

Step 5: Choose the Correct Sign

To ensure that \( e^y > 0 \) (since it is an exponential term), we take the positive sign in \( e^y = x \pm \sqrt{x^2 + 1} \). If \( x + \sqrt{x^2 + 1} \) were negative, it would contradict the fundamental property of exponentials being positive, hence we use "\(+\)".

Step 6: Solve for y Using the Natural Logarithm

To solve for \( y \), take the natural logarithm of both sides: \[ y = \ln(e^y) = \ln(x + \sqrt{x^2 + 1}) \]. Since \( y \) was defined as \( \sinh^{-1}(x) \), it follows that: \[ \sinh^{-1}(x) = \ln(x + \sqrt{x^2 + 1}). \] Thus, the derivation is complete.

Key Concepts

Hyperbolic Functions

Hyperbolic functions are analogs of trigonometric functions that arise from hyperbolas instead of circles. These functions include hyperbolic sine (\(\sinh(x)\)), hyperbolic cosine (\(\cosh(x)\)), and hyperbolic tangent (\(\tanh(x)\)), among others. A key property is their connection to exponential functions, evident from their definitions:

• \(\sinh(x) = \frac{e^x - e^{-x}}{2}\)
• \(\cosh(x) = \frac{e^x + e^{-x}}{2}\)

These functions are useful in many areas, including calculus, among others. Unlike trigonometric functions, hyperbolic functions do not have periodicity. They instead grow indefinitely or approach a limit as their input increases. Understanding how they relate to the exponential function is beneficial for deeper mathematical analysis.

Logarithmic Functions

Logarithmic functions are the inverses of exponential functions. The natural logarithm, denoted as \(\ln(x)\), is particularly important, where its base is 'e' (an irrational constant approximately equal to 2.71828). When we say that \(\ln(b) = a\), it means that \(e^a = b\).
Logarithmic functions can transform multiplicative relationships into additive ones. This property proves highly useful in various fields like physics and engineering. Understanding the inverse nature of logarithms helps in solving equations involving exponentials easily. In the derivation of inverse hyperbolic functions, the natural logarithm plays a vital role. It helps translate an exponential form back to an algebraic expression, as seen in transforming \(e^y\) back to \(y\) using \(\ln\).

Quadratic Equation

Quadratic equations are polynomials of degree two in the form \(ax^2 + bx + c = 0\). Solutions for these equations can be found using the quadratic formula:

• \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\)

This formula is derived from completing the square process. In the context of inverse hyperbolic functions, solving a quadratic equation helps to express \(e^y\) as \(x \pm \sqrt{x^2 + 1}\)
Quadratics often arise in physics, economics, and different scientific computations. Understanding how to use this formula is crucial for solving such equations, providing roots that are critical points in various functions.

Exponential Functions

Exponential functions are characterized by a constant base raised to a variable exponent, usually written as \(f(x) = a^x\). The most significant exponential function is the natural exponential function \(e^x\), where 'e' is Euler's number. These functions naturally model growth and decay processes due to their property of compounding:

• When the base is greater than 1, they show growth (such as population growth).
• When the base is between 0 and 1, they show decay (such as radioactive decay).

Their properties make exponential functions fundamental in fields such as finance, sciences, and technology. In the derivation process of inverse hyperbolic functions, manipulating exponentials is pivotal in showing the relationship and constructing the desired function expression. Understanding how pairs like \(e^x\) and its inverse, \(\ln(x)\), are tightly related allows for simplifying complex mathematical expressions.