Mathematics

Problem

Consider

\displaystyle \int_{-\infty}^{3} \dfrac{1}{(x-1)^{2}}\,dx

. Which of the following statements is accurate? a) The integral converges because

\dfrac{1}{(x-1)^{2}}

is a

p

-integrand with

p=2>1

, so the area is finite. b) The integral diverges because the integrand has a non–integrable

p

-type singularity at

x=1

; the limits from both sides of

1

blow up to

+\infty

, so the improper integral fails to converge. c) The integral diverges solely because of the infinite interval

(-\infty,1)

; the singularity at

x=1

is integrable, so that part is harmless. d) The integral converges to

1

because the infinite area near

x=1

exactly cancels the infinite tail extending to

-\infty