Problem 55 - Calculus for Biology and Medicine

Question

In Problems 43-58, use substitution to evaluate each definite integral. $$ \int_{e}^{e^{2}} \frac{d x}{x(\ln x)^{2}} $$

Step-by-Step Solution

Answer

The value of the integral is \( \frac{1}{2} \).

Step 1: Identify Substitution Variable

To evaluate the integral \( \int_{e}^{e^{2}} \frac{d x}{x(\ln x)^{2}} \) using substitution, we first choose a substitution. We notice that \( \ln x \) is a likely candidate for substitution since its derivative relates to the integrand. Let \( u = \ln x \).

Step 2: Compute Differential and Bounds of Integration

Differentiate the substitution: \( du = \frac{1}{x} \, dx \). This matches the \( \frac{1}{x} \, dx \) in our integrand. Next, change the bounds of integration: When \( x = e \), \( u = \ln e = 1 \). When \( x = e^2 \), \( u = \ln e^2 = 2 \).

Step 3: Substitute Variables and Change the Integral

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

Step 4: Evaluate the New Integral

The integral \( \int \frac{1}{u^2} \, du \) is equal to \( -\frac{1}{u} + C \) after applying the power rule for integration. Applying this gives: \[ \int_{1}^{2} \frac{1}{u^2} \, du = \left[ -\frac{1}{u} \right]_{1}^{2} \].

Step 5: Compute and Simplify the Definite Integral

Evaluate \( \left[ -\frac{1}{u} \right]_{1}^{2} \). This is \( \left( -\frac{1}{2} \right) - \left( -1 \right) = \frac{1}{2} \).

Key Concepts

Substitution Method

The substitution method is a powerful technique in calculus for simplifying the process of integration. It involves changing variables to make an integral easier to solve. In essence, you substitute part of the integrand (the function you're integrating) with a new variable. This new variable, when appropriately chosen, will often make the integration process much more straightforward.

For example, if you encounter an integral such as \( \int \frac{d x}{x(\ln x)^{2}} \), identifying parts of the integrand that could be replaced using a substitution is crucial. In this case, noticing \( \ln x \) as part of the integrand prompts the substitution \( u = \ln x \), which simplifies subsequent steps.

• This method can be particularly useful when dealing with complicated functions or compositions of functions.
• It leverages derivatives, as choosing an effective substitution often involves recognizing the derivative of a portion of the integrand.'
• After making a substitution, it is also important to adjust the limits of integration to match the new variable, if dealing with definite integrals.

Understanding the substitution method opens up possibilities for tackling a wider range of integral problems with ease and efficiency.

Integration by Substitution

Integration by substitution is akin to reversing the chain rule from differentiation. By choosing a part of the integral to substitute, you reframe the problem in simpler terms. This transformation from one variable to another is crucial for evaluating tougher integrals.

In the given problem, \( \int \frac{d x}{x(\ln x)^{2}} \), converting it via substitution transforms the integral into a simpler one. With \( u = \ln x \), we then have \( du = \frac{1}{x} \, dx \). This substitution means that our integral becomes \( \int \frac{1}{u^2} \; du \), a simpler expression to integrate.

• Integration by substitution reduces complexity by converting complex functions into simple polynomial integrals.
• Following the substitution, a straightforward computation yields the antiderivative, and if a definite integral, requires reevaluating limits in terms of the new variable.
• It relies on identifying derivatives within the integral that align with substitution choices, turning integration into a manageable task.

This method's effectiveness underscores why it's a staple of integration techniques, offering a means to handle otherwise difficult integrals effectively.

Calculus Integration Techniques

The realm of calculus integration techniques is vast, including a variety of methods designed to tackle different types of integrals. While substitution is a prevalent approach, there are others like integration by parts, partial fraction decomposition, and trigonometric integrals intended for specific forms of functions.

Substitution specifically is a versatile tool, particularly in problems where there’s an apparent function derivative present or transformable in the integrand. Successfully identifying parts to substitute can drastically simplify an integral, as seen with transforming \( \int \frac{d x}{x(\ln x)^{2}} \) to a manageable polynomial integral using substitution.

• Understanding when and how to apply substitution often requires a practiced eye to notice derivative patterns or potential transformations.
• Combining substitution methods with other integration techniques as needed can solve very intricate integrals.
• Grasping a broad array of techniques equips you with a toolkit to solve integrals across contexts, adapting strategies as required by the function type.

Whether facing engineering, physics, or pure math problems, mastering these calculus techniques enhances problem-solving capabilities and mathematical fluency.