CPM Homework Help : CALC Problem 6-168

Determine what makes the integrals below "improper." Then, determine how to rewrite the integral using "proper" limit notation. Finally, find the area represented by each.

$\int _ { 0 } ^ { \infty } e ^ { - x } d x$

$\int _ { - 3 } ^ { 2 } \frac { 1 } { 2 x + 6 } d x$

$\int _ { 1 } ^ { \infty } \frac { \operatorname { ln } x } { x } d x$

Hint:

Hint (a):

Does the graph of the integrand, y = e⁻^x, have any asymptotes on the bounded domain, [0, ∞)?

Steps (a):

$\lim\limits_{a\to \infty }\int_{0}^{a}e^{-x}dx=\lim\limits_{a\to \infty }-e^{-x}\Big|_0^a=\underline{\ \ \ \ \ \ \ }$

Answer (a):

The answer should be very pleasing.

Hint (b):

$\text{Does the graph of the integrand, }y= \frac{1}{2x-6},$ have any asymptotes on the bounded domain, [–3, 2]?

Hint (c):

Before you rewrite as a limit, use U-substitution on the integrand and the bounds. Let U = ln(x).