Problem 2 - College Algebra and Calculus: An Applied Approach

Question

Decide whether the integral is improper. Explain your reasoning. $$ \int_{1}^{3} \frac{d x}{x^{2}} $$

Step-by-Step Solution

Answer

No, the integral is not improper. The integral is not improper because both the interval of integration (which is [1,3]) and the function (which is \( \frac{1}{{x^{2}}} \)) are bounded.

Step 1: Identify the Interval of Integration

The interval of integration is [1,3]. This is a finite or bounded interval. So, the interval of integration is not making the integral improper.

Step 2: Check for Function's Boundedness within the Interval

The function to be integrated is \( \frac{1}{{x^{2}}} \). As x varies from 1 to 3, the function is well defined; it has no discontinuities within the interval and does not tend to infinity. Hence, the function is bounded within the interval [1,3].

Step 3: Conclude about the Improperness of the Integral

Since both the interval of integration is bounded and the function is bounded within this interval, the given integral is not considered as an improper integral. An integral can only be improper if either the interval of integration is unbounded (which is not the case here) or the function to be integrated is unbounded within the interval of integration (which is not the case here, too).