Problem 55 - Calculus of a Single Variable

Question

In Exercises 55–60, evaluate the integral. $$ \int_{0}^{\ln 2} \tanh x d x $$

Step-by-Step Solution

Answer

\(\ln(\frac{3}{2})\)

Step 1: Identify and confirm the basic integral form

The integral is in the standard form \(\int \tanh x\, dx\). The antiderivative of this form, which can be found in integral tables, is known to be \(\ln(\cosh x)\).

Step 2: Evaluate the antiderivative at the upper and lower limits of integration

Using the antiderivative from step 1, apply the fundamental theorem of calculus and evaluate the antiderivative at \(\ln 2\) and 0. The difference, \(\ln(\cosh(\ln 2)) - \ln(\cosh(0))\), will give the value of the integral.

Step 3: Simplify the solution

Simplify this subtraction. Here, \(\cosh(\ln 2)=\frac{1}{2}e^{ln 2}+ \frac{1}{2}e^{-ln 2}= \frac{3}{2}\), and \(\cosh(0)=1\). Thus, the integral evaluates to \(\ln(\frac{3}{2}) - \ln(1)\). And since \(\ln(1) = 0\), the solution simplifies to \(\ln(\frac{3}{2})\).

Key Concepts

Antiderivative

Understanding antiderivatives is essential when evaluating integrals. An antiderivative of a function is another function such that when it is differentiated, the original function is obtained. For instance, if we have a function denoted by \( f(x) \), its antiderivative is a function \( F(x) \) such that \( F'(x) = f(x) \).

Regarding the exercise, the antiderivative of the hyperbolic tangent function \( \tanh x \) is known and can be found in most integral tables as \( \ln(\cosh x) \). The ability to recognize and apply the antiderivative of a function accurately is a pivotal step in solving integrals and requires familiarity with a range of fundamental antiderivatives. To improve students' understanding, it is beneficial to practice finding antiderivatives of various functions, building a foundation that will make evaluating integrals more straightforward.

Fundamental Theorem of Calculus

The Fundamental Theorem of Calculus bridges the concepts of differentiation and integration and plays a critical role in evaluating definite integrals. It comes in two parts: the first part establishes the relationship between differentiation and integration, and the second part, which we apply here, provides a method to evaluate definite integrals.

According to the second part, if \( F(x) \) is an antiderivative of \( f(x) \) over an interval [a, b], then the definite integral of \( f(x) \) from \( a \) to \( b \) is simply \( F(b) - F(a) \). This powerful theorem simplifies the process by eliminating the need to sum infinite rectangles under the curve, as traditionally done in Riemann sums. Instead, one only needs to find the antiderivative and evaluate it at the bounds of the integral. In our exercise, by applying this theorem, students can solve the given integral by finding the antiderivative of \( \tanh x \) and then evaluating it at \( \ln 2 \) and \( 0 \).

Hyperbolic Functions

Hyperbolic functions, including \( \sinh x \), \( \cosh x \), and \( \tanh x \), are analogs of the trigonometric functions but for a hyperbola rather than a circle. Just as trigonometric functions are vital for circular and oscillatory motion, hyperbolic functions are important in many areas of mathematics, including calculus, complex analysis, and hyperbolic geometry.

These functions are defined using exponential functions. For example, \( \cosh x = (e^x + e^{-x})/2 \) and \( \tanh x = \sinh x / \cosh x = (e^x - e^{-x})/(e^x + e^{-x}) \). Understanding their properties, such as their derivatives and antiderivatives, is crucial for solving integrals that involve hyperbolic functions. For the exercise in question, knowing that the antiderivative of \( \tanh x \) is \( \ln(\cosh x) \) allows for direct evaluation of the integral. Learning how to derive and work with hyperbolic functions equips students with tools to tackle problems in advanced mathematics and physics.