Question

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

The integral \(\int_{0}^{1} \frac{2 x-5}{x^{2}-5 x+6} dx\) is not improper because the function being integrated is not undefined at any point within the interval of integration [0,1] and both limits of integration are finite.

Step-by-step solution

Identify values where denominator is zero

Firstly solve the equation of denominator equal to zero, \(x^{2}-5 x+6 = 0\). By applying factor decomposition, we get \((x - 2)(x - 3) = 0\). So, x can be 2 or 3.

Check if these values are in the interval of integration

The values 2 and 3 are not in the integral's interval [0,1]. So, there is no point within the interval of integration at which the function being integrated is undefined.

Check if limits are infinity

The limits of integration, 0 and 1, are both finite. The function is defined at these points, which means there is no issue regarding them.