Problem 45 - College Algebra and Calculus: An Applied Approach

Question

Evaluate the definite integral. $$ \int_{0}^{2} \frac{x}{1+4 x^{2}} d x $$

Step-by-Step Solution

Answer

The value of the definite integral is \( \frac{1}{8}\ln 17 \).

Step 1: Choose a suitable substitution

Let \( u = 1+4x^2 \). The derivative \( du = 8xdx \), or rearranging, \( dx = \frac{du}{8x} \). Also, when \( x = 0 \), \( u = 1 \), and when \( x = 2 \), \( u = 1+4(2^2)=17 \).

Step 2: Substitute into the integral

The integral becomes \( \int_{1}^{17} \frac{x}{u} \cdot \frac{du}{8x}\). The factor of \( x \) cancels out, so we get \( \frac{1}{8}\int_{1}^{17} \frac{1}{u} du\).

Step 3: Compute the antiderivative

The antiderivative of \( \frac{1}{u} \) is \( \ln|u|\), so \( \frac{1}{8}\int_{1}^{17} \frac{1}{u} du= \frac{1}{8} \left[ \ln|17| - \ln|1| \right]\).

Step 4: Simplify the result

Simplify the antiderivative to obtain \( \frac{1}{8}(\ln 17 - \ln 1)=\frac{1}{8}\ln 17 \).

Key Concepts

Understanding the Substitution Method

The substitution method is a handy tool for solving integrals. It helps in transforming a difficult integral into a simpler one. In this method, we choose a substitution, often denoted by \( u \), to simplify the integral's structure.
In our exercise, we used the substitution \( u = 1 + 4x^2 \). This choice is strategic because it encapsulates the more complicated part of the integrand:

• It simplifies the denominator without leaving any complex terms in the expression.
• The derivative of the substitution, \( du = 8x \, dx \), is related to the \( x \, dx \) term, allowing these terms to cancel out effectively.

Understanding substitution goes beyond reaching a solution. It’s about identifying the most convenient substitution to make integration feasible. It requires us to think about how the expression behaves, both in its current and simplified forms.

Grasping the Concept of Antiderivative

An antiderivative provides a function whose derivative results back to the original function. This is a core idea when dealing with integrals. The definite integral represents an accumulated value over an interval, and its computation leans heavily on finding the antiderivative.
In this exercise, once the substitution was made, the integral transformed to \( \frac{1}{8} \int_{1}^{17} \frac{1}{u} \, du \). Here, we face a standard form whose antiderivative we should recognize: \( \int \frac{1}{u} \, du \) has a straightforward antiderivative, \( \ln|u| \).
Computing the antiderivative involves:

• Recognizing the integral’s format and matching it with known antiderivatives.
• Ensuring the entire integral's limits and constants align before evaluation.

Smoothly transitioning from the integral to its antiderivative replaces complexity with a direct path to computation.

The Role and Rules of Logarithmic Functions

Logarithmic functions play a powerful role in calculus, especially when working with antiderivatives and integrals. The standard logarithmic function, denoted as \( \ln(x) \), is the natural logarithm base \( e \). In integration, it emerges as the antiderivative of rational functions like \( \frac{1}{x} \).
In our solved integral, \( \frac{1}{8} \left[ \ln|17| - \ln|1| \right] \), the logarithmic function becomes crucial. It simplifies the antiderivative of \( \frac{1}{u} \) into a manageable form.
Keep in mind the properties of logarithms that assist in simplifying expressions:

• The rule \( \ln \frac{a}{b} = \ln a - \ln b \), helps in reducing the evaluation of definite integrals' limits.
• Knowing that \( \ln|1| = 0 \), provides quick simplifications where applicable.

Understanding these properties of logarithmic functions can significantly streamline solutions, especially when tackling integrals and their antiderivatives.