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Chapter 6: Problem 3

Decide whether the integral is improper. Explain your reasoning. $$ \int_{0}^{1} \frac{2 x-5}{x^{2}-5 x+6} d x $$

Short Answer

Expert verified
The integral is not improper. It is a proper integral because the integrand (the function to be integrated) does not have any points within the interval [0,1] where it is undefined or has unbounded behaviour. Hence, the integral can be evaluated with standard techniques.

Step by step solution

01

Check denominator for possible zero values

The denominator of the given fraction is \(x^{2}-5x+6\). To check when it could be zero, solve the equation \(x^{2}-5x+6=0\). By factoring, the equation becomes \((x-2)(x-3)=0\). So, the denominator is zero when x=2 or x=3. Now, we need to check if any of these values lie within the provided range.
02

Compare the roots with the limits of the integral

The roots are x=2 and x=3, but the limits of the integral given are from 0 to 1. Notably, these values (2 and 3) are not within this range (0,1). Therefore, there are no points within the interval [0,1] that make the denominator zero.
03

Check for unbounded behavior of the integrand

Rational functions can be unbounded at the points where the denominator is zero. However, we've already found that such points, if any, do not exist within our integration limits. So, it's safe to say that the integrand has no unbounded behaviour within the interval [0,1].

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