Question

Evaluate the following integrals or state that they diverge. $$\int_{-1}^{1} \frac{x}{x^{2}+2 x+1} d x$$

Short answer

Answer: The integral diverges.

Step-by-step solution

Identify the function and the interval

The function we need to evaluate is given by: $$f(x) = \frac{x}{x^{2}+2x+1}$$ And we are given the interval: $$[-1, 1]$$

Analyze the function for any divergence

To check for divergence, we first need to simplify the denominator and see if the function is undefined at any point within the interval. The denominator can be simplified as: $$x^{2} + 2x + 1 = (x+1)^{2}$$ So the original function can be rewritten as: $$f(x) = \frac{x}{(x+1)^{2}}$$ The function is undefined when the denominator is equal to 0, which occurs if \(x = -1\). Since \(-1\) is within the given interval, the integral diverges.

State the result

Since the integral contains a point of divergence within the interval, we can conclude that: $$\int_{-1}^{1} \frac{x}{x^{2}+2 x+1} d x$$ Diverges.

Key concepts

Understanding Divergence in Improper Integrals

Improper integrals can sometimes diverge, meaning they do not have a finite value. Divergence occurs when an integral has a point where the function is not defined or the values are not finite within the interval of integration. In the case of the integral \( \int_{-1}^{1} \frac{x}{x^{2}+2x+1} \, dx \), we notice a point of interest at \( x = -1 \). This is because the function becomes undefined at this point - the denominator turns into zero, creating an infinite value at \( x = -1 \).
The whole idea of checking for divergence is to identify these undefined or infinite points. If there are any present within the interval of integration, like our interval of \([-1, 1]\), it is concluded that the integral diverges and does not solve to a real number.

• Divergence can be identified through undefined points.
• The denominator turning zero within the interval indicates divergence.
• An improper integral that diverges cannot be evaluated to a finite number.

Exploring Intervals of Integration

The interval of integration is the range over which the function is evaluated in an integral. For our problem, the interval is \([-1, 1]\). Understanding this concept is essential because any improper behavior within the interval, such as division by zero, could impact the integral's existence.
The evaluation of an integral heavily relies upon the values and behavior of the function across the entire interval. If there is a point in the interval where the function is not defined, it will affect the convergence of the integral. That's exactly what occurs at \(x = -1\), where the denominator becomes zero.

• Intervals define the range for integration.
• Problems in the interval can lead to divergence.
• Analyze endpoints and points within effectively for undefined behavior.

Simplifying Functions for Better Analysis

Function simplification is a powerful tool to understand how an integral behaves. Before attempting to evaluate or determine if an integral diverges, it's crucial to look at the function in a simplified form.
This simplification can involve factoring polynomials or reducing fractions. In our given function \(f(x) = \frac{x}{x^{2}+2x+1}\), simplification reveals \(f(x) = \frac{x}{(x+1)^{2}}\). This reveals the root problem at \(x = -1\), where the denominator becomes zero, thus indicating divergence.

• Simplification makes spotting issues easier.
• Rewrite functions to understand potential problems.
• Factor denominators to find crucial undefined points.

By simplifying, we not only make the integral easier to work with, but we also unveil the hidden points of divergence effectively.