Integration by parts and two improper…

Name: ______________________ Date: ____________ Integration by parts and two improper integrals (9 points)

Graph

Let

f

be the function defined by

f(x) = (2x + 1)e^{-x}

. The graph of

f

for

x \ge 0

is shown. The region

R

lies under the graph of

f

, above the

x

-axis and to the right of the

y

-axis, and goes on to the right without end. A calculator is not allowed.

(a) Use integration by parts to find the value of $\int_0^1 (2x + 1)e^{-x}\,dx$ . Show the work that leads to your answer. [3 points]
(b) Determine whether the area of $R$ is finite. If it is, find it; if it is not, explain why. Use correct limit notation throughout. [4 points]
(c) Determine whether $\int_1^\infty \frac{2x}{x^2 + 1}\,dx$ converges or diverges. Show the work that leads to your answer. [2 points]

Integration by parts and two improper integrals