Problem 59 - Calculus: Early Transcendental Functions

Question

There are often multiple ways of computing an antiderivative. For \(\int \frac{1}{x \ln \sqrt{x}} d x,\) first use the substitution \(u=\ln \sqrt{x}\) to find the indefinite integral \(2 \ln |\ln \sqrt{x}|+c .\) Then rewrite \(\ln \sqrt{x}\) and use the substitution \(u=\ln x\) to find the indefinite integral 2 In \(|\ln x|+c .\) Show that these two answers are equivalent.

Step-by-Step Solution

Answer

The two different indefinite integrals \(2 \ln |\ln \sqrt{x}| + c\) and \(2 \ln |\ln x| + c\) are equivalent as the natural logarithm is invariant under the square root.

Step 1 - First Substitution to Solve Integral

First, let us find the antiderivative using the substitution \(u = \ln \sqrt{x}\). In this case, \(du = \frac{1}{2x}dx\). Then, the integral becomes \(\int \frac{du}{2u}\) which gives the indefinite integral \(2 \ln |u| + c = 2 \ln |\ln \sqrt{x}| + c\).

Step 2 - Second Substitution to Solve Integral

Now, we rewrite \(\ln \sqrt{x}\) as \(\frac{1}{2} \ln x\). Then use the substitution \(v = \ln x\). With this substitution, \(dv = \frac{1}{x}dx\). Hence, the integral now becomes \(\int \frac{dv}{2v}\). This results in the new indefinite integral \(2 \ln |v| + c = 2 \ln |\ln x| + c\).

Step 3 - Proving Equivalence

Finally we show that the two answers are equivalent. We know that the natural logarithm is invariant under the square root, i.e., \(\ln \sqrt{x} = \frac{1}{2} \ln x\). So, it follows that \(\ln |\ln \sqrt{x}| = \ln |\frac{1}{2} \ln x| = \ln |\ln x|\), hence both answers are equivalent.

Key Concepts

Indefinite Integral

An indefinite integral, or an antiderivative, is a fundamental concept in calculus that represents the reverse of differentiation. For a function f(x), an indefinite integral of f with respect to x is a function F(x) such that F'(x) = f(x).

The notation for an indefinite integral is \( \int f(x) \, dx \), and the general solution is written as \( F(x) + C \), where \( C \)) is the constant of integration. This constant represents all possible values that could be added to the function F due to the differential operator losing information about shifts along the y-axis.

A typical step in calculating an indefinite integral is to simplify the function or to use a method, such as substitution, to make the integral more manageable to solve. The final result should be a function that, when differentiated, gives back the original function f(x).

Substitution Method

The substitution method is a technique used to simplify the process of finding indefinite integrals, especially when an integral is not straightforward. This method involves substituting part of the integral with a new variable that simplifies the integral into a more familiar form.

To apply the substitution method, you need to identify a section of the integral that when replaced, will enable easier integration. Let's take the given exercise as an example. After substituting \( u = \ln \sqrt{x} \), the integral becomes less complicated as we integrate with respect to a new variable \( u \), making it easier to find the antiderivative in terms of \( u \). After integrating, we back-substitute \( u \)) with its original expression to return to the variable \( x \), providing a solution in terms of the original variable.

This method not only simplifies integration but also opens the door to handling more complex integrals that would be difficult or impossible to solve directly.

Natural Logarithm

The natural logarithm is a logarithm with base \( e \) where \( e \) is an irrational and transcendental number approximately equal to 2.71828. The natural logarithm of a number \( x \) is often denoted as \( \ln(x) \) and is the power to which \( e \) must be raised to obtain the number \( x \).

One important property that the natural logarithm holds is that \( \ln(a^b) = b \cdot \ln(a) \). This property was used in the exercise to rewrite \( \ln \sqrt{x} \) as \( \frac{1}{2} \ln x \) before applying the second substitution. Natural logarithms play a crucial role in the calculus of exponential growth and decay processes and are extensively utilized in solving integration problems involving exponential functions.

It is also important to note that the natural logarithm function is continuous and differentiable over its entire domain except at zero, and its inverse function is the exponential function \( e^x \). Understanding the behavior and properties of natural logarithms can help in solving complex integrals and in various applications across applied mathematics, physics, and engineering.