Question

In Exercises \(47-52,\) evaluate the integral. \(\int_{0}^{\ln 2} \tanh x d x\)

Short answer

The evaluation of the integral \int_{0}^{\ln 2} \tanh x d x is \(\ln 2\).

Step-by-step solution

Understand the integral

The integral is asking for the area under the curve of the function \(\tanh(x)\) from 0 to \(\ln 2\). Hyperbolic functions have unique properties that will help in solving this problem.

Change the form of the integral

The hyperbolic tangent function can also be written as the quotient of the hyperbolic sine and the hyperbolic cosine: \(\tanh x = \frac{\sinh x}{\cosh x}\). Therefore, the integral becomes: \(\int_{0}^{\ln 2} \frac{\sinh x}{\cosh x} dx\)

Use the standard substitution method

The substitution method is used to simplify and solve integral problems. Here, set \(u = \cosh x\), then \(du = \sinh x dx\). Therefore, the new integral in terms of \(u\) is \(\int \frac{1}{u} du\)

Solve the new integral

This integral is a standard integral and its solution is \(\ln |u|\). Replace \(u\) with \(\cosh x\) to get \(\ln |\cosh x|\).

Apply the limits of the definite integral

Apply the limits of the integral from 0 to \(\ln 2\) to the solution of the integral obtained in the previous step, it results in \(\ln |\cosh(\ln 2)| - \ln |\cosh(0)|\).

Calculate and simplify

Since \(\cosh x = \frac{e^x + e^{-x}}{2}\) and \(\cosh 0 = 1\), this gives \(\ln |\frac{2 + 1/2}{2}| - \ln |1| = \ln 2 - 0\)

Key concepts

Hyperbolic Functions

Hyperbolic functions are analogues to the ordinary trigonometric functions but are based on hyperbolas instead of circles. Specifically, the hyperbolic tangent function, \(\tanh x\), which we encounter in the given exercise, is defined as the ratio of the hyperbolic sine function \(\sinh x\) to the hyperbolic cosine function \(\cosh x\), that is \(\tanh x = \frac{\sinh x}{\cosh x}\).

Both \(\sinh x\) and \(\cosh x\) can be expressed in terms of exponential functions: \(\sinh x = \frac{e^x - e^{-x}}{2}\) and \(\cosh x = \frac{e^x + e^{-x}}{2}\). This makes hyperbolic functions particularly well-suited for certain calculus operations, including integration, because their derivatives are relatively simple. Moreover, hyperbolic functions exhibit certain properties useful in mathematical analysis, such as \(\cosh^2 x - \sinh^2 x = 1\), which is analogous to the Pythagorean identity for trigonometric functions.

Substitution Method for Integrals

The substitution method for integrals, also known as u-substitution, is a technique to simplify an integral by changing the variable of integration. This is particularly helpful when dealing with composite functions or functions with complicated expressions. The method involves choosing a new variable, frequently denoted as \(u\), which represents part of the integrand's expression.

To apply this technique, we determine \(u\), and then find its derivative \(du\). Our goal is to rewrite the original integral in terms of \(u\) entirely, thereby simplifying it to a form that is easier to evaluate. In this exercise, by setting \(u = \cosh x\), we could rewrite the integral in terms of \(u\), since the derivative of \(\cosh x\) is \(\sinh x\), which matches part of the integrand, enabling us to use this method effectively.

Natural Logarithm Properties

The natural logarithm, denoted as \(\ln\), has unique properties that make it an invaluable tool in calculus. One of the key properties used in evaluating definite integrals is that the natural logarithm of a product is equal to the sum of the natural logarithms of the individual factors, while the natural logarithm of a quotient is the difference of the natural logarithms of the numerator and the denominator. More specifically, we have: \(\ln(ab) = \ln a + \ln b\) and \(\ln(\frac{a}{b}) = \ln a - \ln b\).

Further, the natural logarithm of 1 is 0, i.e., \(\ln 1 = 0\). In the context of the given problem, these properties allow us to simplify the integral's solution from Step 6 by demonstrating that \(\ln |\cosh(0)| = \ln |1| = 0\), further reducing the expression to \(\ln 2\). These logarithmic identities are crucial in solving various mathematical problems, particularly when integrating functions that involve the natural logarithm.