Problem 355 - Calculus Volume 1

Question

In the following exercises, evaluate the definite integral. $$\int_{1}^{2} \frac{1+2 x+x^{2}}{3 x+3 x^{2}+x^{3}} d x$$

Step-by-Step Solution

Answer

The integral evaluates to \( \ln(2) \).

Step 1: Simplify the Integrand

First, let's consider the expression inside the integral: \( \frac{1+2x+x^2}{3x+3x^2+x^3} \). Notice that the numerator \( 1+2x+x^2 \) can be rewritten as \( (x+1)^2 \), and the denominator \( 3x+3x^2+x^3 \) can be rewritten as \( x(x+1)^2 \). The integrand simplifies to \( \frac{(x+1)^2}{x(x+1)^2} = \frac{1}{x} \).

Step 2: Set Up the Integral with the Simplified Function

After simplifying the integrand, our integral becomes \( \int_{1}^{2} \frac{1}{x} \, dx \). This is much simpler to evaluate.

Step 3: Evaluate the Integral

The integral of \( \frac{1}{x} \) is the natural logarithm function: \( \ln|x| \). Therefore, the definite integral from 1 to 2 is evaluated as:\[ \int_{1}^{2} \frac{1}{x} \, dx = \ln|x| \bigg|_{1}^{2} = \ln(2) - \ln(1) \].

Step 4: Simplify the Result

We know that \( \ln(1) = 0 \), so the expression simplifies to \( \ln(2) \). Thus, the value of the definite integral is \( \ln(2) \).

Key Concepts

Integral Simplification

Simplifying an integral can make the solving process much easier and more efficient. To simplify, we need to look at the expression within the integral. The goal is to make it as simple as possible.

In our example, the integrand is \( \frac{1+2x+x^2}{3x+3x^2+x^3} \). The numerator \( 1+2x+x^2 \) can be rewritten using algebraic identities. It becomes \( (x+1)^2 \). Similarly, the denominator \( 3x+3x^2+x^3 \) can also be simplified to \( x(x+1)^2 \).

When both the numerator and the denominator have a common factor, in this case \( (x+1)^2 \), they can be canceled out. That's how the integrand reduces to \( \frac{1}{x} \). This step is crucial because it leads to a simpler expression that is much easier to integrate.

Natural Logarithm

Once the integrand is reduced to \( \frac{1}{x} \), the task is to find its integral. The integral of \( \frac{1}{x} \) is one of the most fundamental integrals, resulting in the natural logarithm function, \( \ln|x| \).

The natural logarithm, usually denoted as \( \ln(x) \), is a logarithm with base \( e \), where \( e \) is an irrational constant approximately equal to 2.71828. It comes up frequently in calculus due to its unique properties, particularly in integrating inverse functions.

Understanding how to work with \( \ln(x) \) is important. Its derivative is \( \frac{1}{x} \), and the natural logarithm itself is used to represent solutions to more complex exponential growth problems. In the integration process, this concept is invaluable as it provides the antiderivative necessary to evaluate the integral.

Evaluating Integrals

Evaluating a definite integral means determining the actual numerical value of the integral over a specific interval. After simplifying the integral to \( \int_{1}^{2} \frac{1}{x} \, dx \), we integrate using \( \ln|x| \).

Here are the steps to evaluate the integral:

• Find the antiderivative, which is \( \ln|x| \).
• Use the limits of integration: substitute \( x=2 \) and \( x=1 \) into \( \ln|x| \).
• Calculate the result: \( \ln(2) - \ln(1) \).

We know that \( \ln(1) = 0 \) because the logarithm of 1 is always zero, for every logarithmic base. Thus, the expression simplifies to just \( \ln(2) \).

Understanding how to compute definite integrals is essential for solving problems in calculus. It provides the means to quantify the total accumulation of a quantity, often used in physics, engineering, and economics to find areas, volumes, and other quantities.