Problem 45 - Thomas Calculus in SI Units

Question

Evaluate the integrals. $$\int_{2}^{4} \frac{d x}{x(\ln x)^{2}}$$

Step-by-Step Solution

Answer

\( \frac{1}{2 \ln 2} \)

Step 1: Recognize the Integral's Form

The given integral is \( \int_{2}^{4} \frac{dx}{x (\ln x)^2} \). This form hints that a u-substitution could simplify the integration process.

Step 2: Choose a Suitable Substitution

To make the integration simpler, let's perform the substitution \( u = \ln x \), which implies \( du = \frac{1}{x} \, dx \), or equivalently, \( dx = x \, du \). Thus, \( x = e^u \).

Step 3: Change the Limits of Integration

Using the substitution \( u = \ln x \), change the limits: when \( x = 2 \), \( u = \ln 2 \); when \( x = 4 \), \( u = \ln 4 \). Hence, the limits change from \( x = 2 \) to \( x = 4 \) into \( u = \ln 2 \) to \( u = \ln 4 \).

Step 4: Substitute and Simplify the Integral

Under the substitution, the integral becomes:\[\int_{\ln 2}^{\ln 4} \frac{1}{(\ln x)^2} \, dx = \int_{\ln 2}^{\ln 4} \frac{x \, du}{x u^2} = \int_{\ln 2}^{\ln 4} \frac{du}{u^2}.\]The \( x \) terms cancel out, reducing the integral to a standard form \( \int \frac{1}{u^2} \, du \).

Step 5: Evaluate the Simplified Integral

To integrate \( \int \frac{1}{u^2} \, du \), rewrite it as \( \int u^{-2} \, du \). The antiderivative is \( -u^{-1} = -\frac{1}{u} \). Therefore, evaluate:\[-\frac{1}{u} \Bigg|_{\ln 2}^{\ln 4}.\]

Step 6: Apply the Limits of Integration

Now substitute the limits \( u = \ln 2 \) and \( u = \ln 4 \) into the antiderivative:\[-\frac{1}{u} \Bigg|_{\ln 2}^{\ln 4} = \left(-\frac{1}{\ln 4}\right) - \left(-\frac{1}{\ln 2}\right) = \frac{1}{\ln 2} - \frac{1}{\ln 4}.\]

Step 7: Simplify the Result

Recognize that \( \ln 4 = 2 \ln 2 \). Therefore:\[\frac{1}{\ln 2} - \frac{1}{\ln 4} = \frac{1}{\ln 2} - \frac{1}{2 \ln 2} = \frac{2}{2 \ln 2} - \frac{1}{2 \ln 2} = \frac{1}{2 \ln 2}.\]This simplifies to \( \frac{1}{2 \ln 2} \).

Key Concepts

Definite Integrals

Definite integrals are a fundamental concept in calculus, providing a way to calculate the area under a curve between two specific points (often referred to as limits). In more intuitive terms, they give us the net accumulation of a quantity over an interval. A definite integral is expressed as \(int_{a}^{b} f(x) \, dx\), where \(a\) and \(b\) are the limits of integration and \(f(x)\) is the integrand.

• The value of a definite integral is a number that represents the total area under the curve of the graph of the function between the limits.
• The process of finding a definite integral involves two main steps: finding the antiderivative (indefinite integral) and then applying the limits of integration.

In our exercise, after substitution and simplification, we evaluated the definite integral on the interval from \( \ln 2 \) to \( \ln 4 \). By calculating the antiderivative and then applying the limits, we found a concise expression for the area under the transformed function.

Natural Logarithm Integration

Integrating functions that involve the natural logarithm \( \ln(x) \) often requires some clever substitution to simplify the process. The natural logarithm, a logarithm with base \(e\), presents unique properties that, when exploited correctly, can streamline integration.

• When you encounter an integral involving \( \ln(x) \), consider using substitution to simplify the expression. For instance, using \( u = \ln x \) can often reduce the complexity.
• In our case, this substitution changed the integrand into the more manageable form \( \int \frac{du}{u^2} \), making it simpler to find the antiderivative.

The substitution transformed a challenging natural logarithm expression into a straightforward integral, allowing us to leverage the elegance of substitution in definite integrals.

Antiderivative

The antiderivative, also known as the indefinite integral, is essentially the reverse of differentiation. It seeks to find a function whose derivative is the given function. This concept is crucial when evaluating integrals because it provides the base for calculating definite integrals.

• In mathematical notation, finding an antiderivative involves the operation \( int f(x) \, dx = F(x) + C \), where \(F(x)\) is the antiderivative and \(C\) is the constant of integration.
• In our example, we found the antiderivative of \( u^{-2} \) to be \( -\frac{1}{u} \).

This antiderivative helped us define the net change represented by the definite integral over its specified limits. Understanding antiderivatives is a core concept to mastering both indefinite and definite integrals.