Calculus II Section 7.7 – Improper Integrals
Recall that the definition of the definite integral

b
a
f ( x) dx requires that [a,b] be finite. Also, the Fundamental
Theorem of Calculus requires that f be continuous on [a,b]. Now we are going to learn to evaluate integrals
where these conditions are not met (i.e., Improper Integrals).
First, consider the integral


1
1
dx . If we extend our idea of definite integrals it makes sense that we would
x2
get the area indicated below.
Does it seem that this area will always be infinite? We will come back to that question later.
Now we will evaluate the integral. We know how to evaluate the integrals given below.


4
1
12
1
1
dx
x2

1
dx
x2


8
1
1
dx
x2
16
1
1
dx
x2
1
dx ? You guessed it! We will take the limit as the upper limit of
x2
 1
b 1
integration approaches infinity. That is,  2 dx  lim  2 dx .
1 x
b  1 x
How can we extend this to evaluate

1
Evaluate:


1
Now evaluate
1
dx
x2


1
1
dx
x
What makes the difference in the value of the integral?
ex
dx
Now evaluate 
 1  e 2 x

The second type of improper integral is the type where there is a point of discontinuity in the interval [a,b]. The
discontinuity could occur at either an endpoint or at a point inside the interval.
Example:

e
0
Now evaluate
ln x 2 dx

1
1
1
dx
x2