Problem 8 - Calculus for Biology and Medicine

Question

Use the Table of Integrals to compute each integral. $$ \int_{e}^{e^{2}} \frac{\ln x}{x} d x $$

Step-by-Step Solution

Answer

The integral evaluates to \(\frac{3}{2}\).

Step 1: Identify the Integration Formula

We need to find an appropriate formula from the Table of Integrals. A closely matching formula is \( \int \frac{\ln x}{x} \, dx = \frac{(\ln x)^2}{2} + C \).

Step 2: Apply the Definite Integral Limits

Since the integral is definite from \(x = e\) to \(x = e^2\), apply the limits: \[\int_{e}^{e^{2}} \frac{\ln x}{x} \, dx = \left[ \frac{(\ln x)^2}{2} \right]_{e}^{e^2}\]

Step 3: Evaluate at the Upper Limit

Substitute \(x = e^2\) into the expression:\[\frac{(\ln e^2)^2}{2} = \frac{(2 \ln e)^2}{2} = \frac{4}{2} = 2\] (using the fact that \(\ln e = 1\)).

Step 4: Evaluate at the Lower Limit

Substitute \(x = e\) into the expression:\[\frac{(\ln e)^2}{2} = \frac{1^2}{2} = \frac{1}{2}\] (since \(\ln e = 1\)).

Step 5: Calculate the Definite Integral

Subtract the lower limit evaluation from the upper limit evaluation:\[\int_{e}^{e^{2}} \frac{\ln x}{x} \, dx = 2 - \frac{1}{2} = \frac{4}{2} - \frac{1}{2} = \frac{3}{2}\]

Key Concepts

integration techniques

When it comes to solving integrals, choosing the correct integration technique can significantly simplify the problem. There are various techniques available, each suited for different kinds of functions and scenarios. One of the most commonly used methods is recognizing patterns and using formulas from a table of integrals, as seen in the given problem. This technique involves recognizing a part of the integrand that matches a known formula, which saves time and avoids the laborious process of deriving a solution from scratch.

• For instance, spotting \( \frac{\ln x}{x} \)in this problem aligns with a ready-made formula \( \int \frac{\ln x}{x} \, dx = \frac{(\ln x)^2}{2} + C \).
• This substitution almost magically transforms a complex-looking integral into an elementary one.

Furthermore, when dealing with definite integrals—those that have specified limits—it is crucial to replace the variable in the antiderivative expression with the upper and lower limits, as shown in this exercise. This application of the limits is straightforward once the correct antiderivative has been identified using the right formula.

logarithmic functions

Logarithmic functions, such as \( \ln x \), are pivotal in calculus and integration problems. They often complement expressions that involve exponential decay or growth.

• The natural logarithm \( \ln x \) is the inverse of the exponential function \( e^x \), making them essential tools for solving equations involving exponentials.
• In this exercise, the function \( \frac{\ln x}{x} \) is integrated, leveraging the properties of logarithmic functions to simplify the integration process.

When integrating, recognizing the inherent properties of logarithms, like \( \ln e = 1 \), significantly simplifies calculations. For example, using known values like \( \ln e^2 = 2 \ln e \) helps in easy calculation of definite integrals, as seen when evaluating at the upper limit in the original solution.

table of integrals

A table of integrals is an invaluable asset for calculus students tackling integration problems. This table includes a variety of pre-established integrals and corresponding antiderivatives. Think of it as a "cheat sheet" or a "reference book" for integration.

• In the exercise provided, a formula from the integral table \( \int \frac{\ln x}{x} \, dx = \frac{(\ln x)^2}{2} + C \) greatly simplifies the evaluation process.
• It avoids the complex and time-consuming process of integration by parts which would typically be needed.

Students should learn to frequently reference these tables, as familiarity can dramatically reduce the time taken to solve integrals, especially during exams or test situations. These tables also serve as great practice tools to understand and memorize common integral forms, enhancing one’s overall integration skills.