Improper Integrals#
Improper Integrals over \([a, \infty)\) and \((-\infty, b]\)#
An improper integral is an integral over an unbounded interval and is defined to be the limit of a definite integral.
Definition
Let \(f\) be a continuous function on the unbounded interval \([a, \infty)\). The improper integral of \(f\) over the interval \([a, \infty)\) is defined by
if the limit exists. Similarly, the improper integral of \(f\) over the interval \((-\infty, b]\) is defined by
if the limit exists.
If an improper integral exists, it corresponds to the area of a region that is unbounded on the left or on the right.
Long Text Description
There is a horizontal x axis with the points 1, 3, 5, 6, and 9 labeled. There is a vertical y axis. The decreasing, concave up curve y = f(x) is plotted on these axes. The region between the curve and the x axis, beginning at x = 1 on the left, is shaded yellow.
Example 1#
Evaluating an improper integral unbounded on the right
Evaluate \(\displaystyle \int_2^\infty \frac{1}{x} ~dx\), if it exists.
Step 1: Evaluate \(\displaystyle \int_2^t \frac{1}{x} ~dx\).
Step 2: Use Step 1 to evaluate \(\displaystyle \int_2^\infty \frac{1}{x} ~dx\) as the limit of a definite integral.
Therefore, the improper integral does not exist.
Example 2#
Evaluating an improper integral unbounded on the left
Evaluate \(\displaystyle \int_{-\infty}^{-1} \frac{1}{x^4}~dx\), if it exists.
Step 1: Evaluate \(\displaystyle \int_s^{-1} \frac{1}{x^4} ~dx\).
Step 2: Use Step 1 to evaluate \(\displaystyle \int_{-\infty}^{-1} \frac{1}{x^4}~dx\) as the limit of a definite integral.
Example 3#
Finding the area under a graph unbounded on the right
Find the area of the region under the graph of \(f(x) = e^{-x/2}\) for \(x \geq 2\).
Step 1: Write the area of the region as an improper integral.
Step 2: Evaluate \(\displaystyle \int_{2}^{t} e^{-x/2}~dx\).
Step 3: Use Step 2 to evaluate \(\displaystyle \int_{2}^{\infty} e^{-x/2}~dx\) as the limit of a definite integral.
Therefore, the area of the region is \(2/e\).
Improper Integrals over \((-\infty, \infty)\)#
Definition
Let \(f\) be a continuous function on \((-\infty,\infty)\). Let \(c\) be a real number, and suppose that the improper integrals
both exist. Then the improper integral of \(f\) over \((-\infty, \infty)\) is defined by
The value of the integral is the same regardless of the value we choose for \(c\) and therefore the value \(c=0\) is typically used.
If an improper integral exists, it corresponds to the area of a region that is unbounded on the left and on the right.
Long Text Description
There is a horizontal x axis with the points -3, -1, 1, 3, 5, and 7 labeled. There is a vertical y axis. The curve y=f(x), which is increasing and concave up as it comes in from the left, becomes increasing and concave down, becomes decreasing and concave down, and then becomes decreasing and concave up as it goes off to the right, is plotted. The entire region between the curve y= f(x) and the x axis is shaded yellow.
Example 4#
Evaluating an improper integral unbounded on both sides
Evaluate \(\displaystyle \int_{-\infty}^\infty \frac{e^x}{(1 + e^x)^3}~dx\), if it exists.
Step 1: Rewrite the integral as a sum of two improper integrals.
Step 2: Compute \(\displaystyle \int \frac{e^x}{(1+e^{x})^3} ~dx\) using the following substitution.
Step 3: Evaluate \(\displaystyle \int_{-\infty}^0 \frac{e^x}{(1 + e^x)^3}~dx\).
Step 4: Evaluate \(\displaystyle \int_0^{\infty} \frac{e^x}{(1 + e^x)^3}~dx\).
Step 5: Evaluate \(\displaystyle \int_{-\infty}^{\infty} \frac{e^x}{(1 + e^x)^3}~dx\).
Example 5#
Evaluate an improper integral unbounded on both sides
Evaluate \(\displaystyle \int_{-\infty}^\infty xe^{-x^2}~dx\), if it exists.