Problem 55 - Calculus for Biology and Medicine

Question

Use substitution to evaluate the definite integrals. $$ \int_{e}^{e^{2}} \frac{d x}{x(\ln x)^{2}} $$

Step-by-Step Solution

Answer

The definite integral evaluates to \(\frac{1}{2}\).

Step 1: Identify Substitution

We need to choose a suitable substitution to simplify evaluating the integral. Notice that the equation involves \( \ln x \). Hence, let's use the substitution \( u = \ln x \). Then, calculate the differential: \( du = \frac{1}{x} \, dx \).

Step 2: Change Limits of Integration

With the substitution \( u = \ln x \), the limits change accordingly. When \( x = e \), \( u = \ln e = 1 \). When \( x = e^2 \), \( u = \ln(e^2) = 2 \). So the new limits of integration are from 1 to 2.

Step 3: Rewrite the Integral

Substitute \( u = \ln x \) and \( du = \frac{1}{x} \, dx \) into the integral. The integral becomes:\[\int_{1}^{2} \frac{1}{u^2} \, du.\]

Step 4: Integrate with Respect to u

The function to integrate now is \( \frac{1}{u^2} \), which is \( u^{-2} \). The integral of \( u^{-2} \) with respect to \( u \) is \( -u^{-1} \). Thus, we integrate and get the expression:\[-\frac{1}{u}\bigg|_1^2.\]

Step 5: Evaluate the Integral

Evaluate the definite integral by substituting the limits:\[-\frac{1}{2} - (-1) = -\frac{1}{2} + 1 = \frac{1}{2}.\]

Key Concepts

Substitution Method

The substitution method is a powerful technique to simplify complex integrals. It's especially useful when direct integration is challenging. In this method, we choose a new variable, typically labeled as `u`, to replace a function or expression in the original integral. This transformation simplifies the integral into a more manageable form. For the exercise, we noticed the presence of \( \ln x \) in the denominator. Therefore, we selected \( u = \ln x \) as our substitution. Once the substitution is made, we also need to express \( dx \) in terms of \( du \). Here, by differentiating \( u \), we find \( du = \frac{1}{x} \, dx \). This correspondence allows the integral to be rewritten purely in terms of \( u \) and \( du \), simplifying the integration process.

Integration Techniques

Integration techniques refer to various strategies used to solve integrals. These include methods like substitution, integration by parts, and partial fractions, among others. Specific to substitution, the goal is to transform the integral into a simpler form that can be easily integrated. After substituting, we often end up with standard integrals with known antiderivatives. In our exercise example, after substituting \( u = \ln x \), the integral simplifies to \( \int \frac{1}{u^2} \, du \). This is a straightforward power rule integration problem. The integral of \( u^{-2} \) is \( -u^{-1} \) or \( -\frac{1}{u} \). These standard antiderivatives are crucial for evaluating definite integrals, and substitution is one of the most helpful integration techniques to reach these forms.

Limits of Integration

In the context of definite integrals, limits of integration define the interval over which the integration occurs. When using substitution, it is important to adjust these limits according to the new variable. Originally, the given integral had limits \( x = e \) to \( x = e^2 \). After our substitution \( u = \ln x \), we must calculate the new limits for \( u \).

• When \( x = e \), \( u = \ln e = 1 \).
• When \( x = e^2 \), \( u = \ln(e^2) = 2 \).

Thus, the integration limits transform to 1 and 2. The revised limits ensure that the region of integration in the \( u \)-domain correctly corresponds to the original \( x \)-domain. When evaluating the final result, always substitute back the new limits into the integrated function and compute the difference. This process confirms that the definite integral retains the same evaluation range, ensuring the calculative accuracy of the integral.