The improper integral is to be analyzed

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

Problem

The improper integral

I=\displaystyle\int_{0}^{\infty} x e^{-x}\,dx

is to be analyzed. Which of the following statements is true? (a) The integral diverges because the factor

x

eventually dominates the exponential decay of

e^{-x}

. (b) The integral converges, but only conditionally; it is not absolutely convergent because the integrand becomes unbounded near

x=0

. (c) The integral converges, but its value is

0

since the integrand equals

0

at both endpoints. (d) The integral converges because

x e^{-x}\to 0

x\to\infty

and the integrand is bounded near

x=0