Consider the integral

MATHEMATICS · Single correct · AP · Calculus BC · Integration and Accumulation of Change · Integration and Accumulation of Change

Problem

\displaystyle \int_{0}^{4}\frac{1}{(x-2)^2}\,dx

. Which statement is true? (a) The integral converges and equals

-1

because

\displaystyle -\frac{1}{x-2}

is finite at both endpoints. (b) The integral converges to

0

since the integrand is symmetric about

x=2

, so the positive and negative areas cancel. (c) The integral diverges because at least one of the two sub-integrals obtained by splitting at

x=2

diverges. (d) The integral is proper because

x=2

lies strictly between

0

and

4

, so the function is continuous at the limits of integration.