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 \(\int_{0}^{1} \frac{2 x-5}{x^{2}-5 x+6} d x\) is not improper, as the function is defined for every value in the range of integration [0,1]. There are no singular points within this interval that would make the integral improper.

Step-by-step solution

Understand the exercise

The exercise is asking to find out if the integral \(\int_{0}^{1} \frac{2 x-5}{x^{2}-5 x+6} d x\) is improper. To do this, the key is to identify if the function has a singular point (where the function is not defined) on the interval [0, 1]. This often occurs when the denominator of a rational function equals zero.

Check the denominator of the rational function

Check the denominator of the rational function for any values that would make it zero. The denominator of the integral function is \(x^{2}-5 x+6\). Setting this equal to zero for solution yields \(x^{2}-5 x+6=0\). Solving this quadratic equation gives the values at which the function is not defined.

Solve the quadratic equation

Solve the quadratic equation \(x^{2}-5x+6=0\) by factoring. Factoring the quadratic expression results in \((x-2)(x-3) = 0\). Setting each factor equal to zero gives two solutions, \(x = 2\) and \(x = 3\). Both of these are outside the interval of integration, [0,1], which means the function is defined for all x in the range of integration.