Question

For the following exercises, find the definite or indefinite integral. $$ \int_{2}^{e} \frac{d x}{x \ln x} $$

Short answer

The integral evaluates to \(-\ln(\ln 2)\).

Step-by-step solution

Identify the form of the integrand

Notice that the integrand is of the form \( \frac{1}{x \ln x} \). This suggests a substitution method might be applicable. Observing that \( \ln x \) appears in the denominator and its derivative \( \frac{1}{x} \) is also present (aside from the constant), indicates a potential substitution.

Choose appropriate substitution

Let \( u = \ln x \), then it follows that \( \frac{du}{dx} = \frac{1}{x} \) or \( du = \frac{1}{x} dx \). This substitution simplifies the integral.

Transform the limits of integration

When \( x = 2 \), \( u = \ln 2 \). When \( x = e \), \( u = \ln e = 1 \). The integral becomes \( \int_{\ln 2}^{1} \frac{1}{u} \, du \).

Integrate using the fundamental formula

The integral \( \int \frac{1}{u} \, du \) is equal to \( \ln |u| + C \). Applying this to the definite integral, we get: \[ \left[ \ln |u| \right]_{\ln 2}^{1} \].

Evaluate the definite integral

Calculate \( \ln |u| \) at the bounds: \[ \ln |1| - \ln |\ln 2| = 0 - \ln (\ln 2) = -\ln (\ln 2). \] Thus, the value of the definite integral is \( -\ln(\ln 2) \).

Key concepts

Substitution Method

In calculus, the substitution method is a useful technique for finding integrals. It simplifies complex integrals by substituting part of the integrand with a single variable, often denoted as \( u \). By choosing \( u \) wisely, we transform the original integral into an easier form. The key is to ensure that the derivative of \( u \) appears in the integrand, allowing us to make a direct substitution.

• First, identify a function within the integrand and its derivative that can be substituted.
• Next, set \( u \) equal to the chosen function, and express \( dx \) in terms of \( du \).
• This method not only simplifies the integrand but also transforms it into a format that's easier to integrate.

In the example provided, we notice that \( \ln x \) is present in the denominator. Since the derivative \( \frac{1}{x} \) is multiplying it, choosing \( u = \ln x \) makes integration straightforward.

Integral Bounds

When dealing with definite integrals, it's crucial to manage the boundaries of integration. These bounds define the interval over which you are finding the area under the curve.
When substitution is applied, these bounds must be adjusted accordingly to reflect the change in variables.

• Initially, identify the original limits of integration, which are the values of the variable at the ends of the interval.
• After substituting, calculate the new limits by evaluating the substitution expression at the original limits.
• This ensures that the new integral remains equivalent to the original, despite the change in variables.

In the exercise, the original bounds are from \( x = 2 \) to \( x = e \). With substitution \( u = \ln x \), they transform to from \( \ln 2 \) to 1.

Natural Logarithm

The natural logarithm, often denoted as \( \ln x \), is a logarithm to the base \( e \), where \( e \) is an irrational constant approximately equal to 2.71828. It is a fundamental function in calculus and often shows up in integration problems, particularly when dealing with exponential growth and decay.
Here's why it's significant in integration:

• \( \ln x \) is the inverse function of the exponential function \( e^x \), providing a bridge between these forms.
• Its derivative is \( \frac{1}{x} \), which is frequently seen in integrands, making it suitable for substitution, as used in the example exercise.
• This makes the integration of \( \frac{1}{x} \ln x \) straightforward upon substitution, as it reduces to a simple form that can be integrated directly.

Indefinite Integral

An indefinite integral is a fundamental concept in calculus representing a family of functions. It involves finding the antiderivative of a given function. Unlike definite integrals, which evaluate to a specific number, indefinite integrals have a general form.
An indefinite integral is denoted as \( \int f(x) \, dx \) and yields a function plus a constant, often written as \( F(x) + C \) where \( C \) is the constant of integration.

• This constant indicates that there are infinitely many antiderivatives that differ by a constant.
• The process of finding an indefinite integral is essentially reversing differentiation.
• This is useful in a variety of applications, including solving differential equations and calculating areas under curves without fixed bounds.

In our scenario, the concept of indefinite integration appeared briefly as the indefinite form \( \ln |u| + C \) before applying the limits of integration.