Question

let ( p ) be a constant. for what value(s) of ( p ) is the integral ( int_{1}^{5} \frac{1}{x - p} dx ) improper? (i.e. for what ( p ) value(s) would we need to use the \improper integrals - type 2\ techniques to solve?) ( p = 1 ) and ( p = 5 ) only. ( p = 0 ) only. this integral is never improper. all ( p ) on the interval ( 1,5 ) only. the integral is improper for all ( p ) on ( (-infty,infty) ).

Explanation:

Step1: Define improper integral condition

An integral is improper if the integrand is undefined at any point in the interval. Here, the integrand $\frac{1}{x-p}$ is undefined when $x=p$.

Step2: Check interval [1,5]

For the integrand to be undefined in [1,5], $p$ must be in [1,5]. So $p=1,2,3,4,5$ make the integral improper.

Answer:

All $p$ on the interval [1,5] only.