Question

Consider the integral \(\int_{0}^{3} \frac{10}{x^{2}-2 x} d x\). To determine the convergence or divergence of the integral, how many improper integrals must be analyzed? What must be true of each of these integrals if the given integral converges?

Short answer

To determine the convergence or divergence of the given integral, two improper integrals (from 0 to 2 and from 2 to 3) must be analyzed. In order for the given integral to converge, both of these integrals must individually converge.

Step-by-step solution

Identify the singularities

Identify the points where the function becomes undefined or infinite. The function is undefined when the denominator equals 0. So the singularities are found by setting \(x^{2} - 2x = 0\), which gives the roots x = 0 and x = 2.

Divide the integral

Given the Integral \(\int_{0}^{3}\frac{10}{x^{2}-2x}\, dx\), it can be divided into two separate integrals at the singularities: \int_{0}^{2}\frac{10}{x^{2}-2x}\, dx + \int_{2}^{3}\frac{10}{x^{2}-2x}\, dx

Check each integral for convergence or divergence

Each of the segregated integrals is then evaluated separately. If both integrals are found to converge, then the given improper integral converges. If either or both diverge, then the given improper integral diverges.

Key concepts

Singularities in Integrals

In calculus, the concept of a singularity refers to points where a function does not behave nicely, that is, it might become infinitely large (approach infinity) or is somehow undefined.

When calculating integrals, these singular points need special attention because they can affect whether the integral is convergent (results in a finite area under the curve) or divergent (where the area is infinite). For instance, the exercise given involves the integral \( \int_{0}^{3} \frac{10}{x^{2}-2 x} dx \), and we must find where the function \( \frac{10}{x^{2}-2 x} \) has singularities.

These singularities emerge when the denominator is zero since division by zero is undefined; consequently, solving the equation \( x^{2} - 2x = 0 \) elucidates the singular points at \( x = 0 \) and \( x = 2 \) for this exercise. These points are crucial because they divide the integral into parts that can be analyzed separately for convergence or divergence.

Improper Integral Analysis

An improper integral is one where either the interval of integration is infinite or the function to be integrated becomes infinite at some points within the integration interval. To analyze such integrals, we first look for singular points or infinite boundaries and then troubleshoot each interval separately.

For the exercise \( \int_{0}^{3} \frac{10}{x^{2}-2 x} dx \), we split the integral at the singular points \( x = 0 \) and \( x = 2 \) to obtain two separate integrals: \( \int_{0}^{2} \frac{10}{x^{2}-2x} dx \) and \( \int_{2}^{3} \frac{10}{x^{2}-2x} dx \). Each part is then assessed for convergence.

This process may involve applying limits where singularity points are included in the intervals, or where an interval is unbounded, to ensure that we are dealing with a well-defined mathematical expression that represents the area correctly.

Convergence Criteria for Integrals

Determining the convergence of an improper integral is crucial for understanding whether the integral has a finite value. There are specific convergence criteria that these integrals must meet.

For instance, to determine if the given improper integral converges, we analyze each subdivided integral around the singularities independently. If all subdivided parts are convergent, the original integral will also be convergent. This involves ensuring that the limit of the integrand exists and is finite as it approaches any singularity point from within the integration range.

For the spotlighted exercise \( \int_{0}^{2} \frac{10}{x^{2}-2x} dx \) and \( \int_{2}^{3} \frac{10}{x^{2}-2x} dx \), the analysis would examine the behavior of each interval as \( x \) approaches the singularity points. If the result of the integral in each interval approaches a finite number, the integral over that interval is said to converge. This process can involve various methods, such as comparison tests, or in certain cases, applying the limit comparison test, to determine whether an integral is convergent or divergent.