Question

Briefly indicate why the following integral is improper. If the integral has multiple reasons for being improper then you must express the integral in terms of integrals that each have only one reason for being improper. For this Checkpoint, you do not need to actually evaluate the integrals. Just write down the integrals that arise. 1. ∫ 0/2x(x−1)x−2 dx

Answer 1

The integral is improper because the function has a singularity at x = 2. It can be split into two integrals, each with its own reason for being improper.

Step-by-step explanation:

An integral is considered improper if one or more of the limits of integration are infinite, or if the integral is undefined at any point within the interval of integration. In this case, the integral ∫ (0/2x)(x−1)(x−2) dx is improper because the function has a singularity at x = 2, which makes the integrand undefined at that point. To express the integral in terms of integrals that each have only one reason for being improper, we can split it into two parts:

• ∫ (0/1)(x−1)(x−2) dx, with the limit of integration from 0 to 1. This integral is improper because it has a singularity at x = 1.
• ∫ (1/2x)(x−1)(x−2) dx, with the limit of integration from 1 to 2. This integral is improper because it has a singularity at x = 2.