The Useful Arctan Integral Form
The following integral is very common in calculus:
1
= arctan + .
1 + A more general form is
1
1
= arctan + .
+
Proof:
Factor from the denominator:
1
1
1
1
=
=
.
+ 1 + ⁄ 1 + ⁄ Now we do a u-du substitution, with = , so that = . Thus, = . We make the
replacements:
1
1
1
1
= .
1 + ⁄ 1 + Note that the a inside the integral comes out to the front, so we have:
1
1
1
1
=
.
1 + 1 + Now we integrate:
1
1
1
1
= arctan = arctan + .
1+
This is an important form! Please memorize it.
Example: Find
5
.
9 + Solution: The 5 can be moved outside, and we have = 9, so that = 3.
have
5
5
= arctan + .
9+
3
3
Thus, using the form, we
Sometimes you need to complete the square:
Example: Find
8
.
+ 4 + 9
Solution: The denominator is + 4 + 9 = + 2 + 5, after completing the square (you should
verify this).
Thus, we now have
8
1
= 8 .
+ 2 + 5
+ 4 + 9
Now it’s in that arctan form. We have = √5 (do you see why?). Therefore, we have
8
1
8
+2
=
arctan #
$ + .
+ 2 + 5
√5
√5
Example: Find
3
.
11 + %
Solution. This requires a u-du substitution: = , so that = 2 . Note that = . Also note
that % = . We make the substitutions:
&
3
3
1
=
#
$.
11 + %
11 + 2
Now simplify. Note that the can be moved outside, and the x’s cancel:
3
1
3
1
#
$
=
.
11 + 2
2 11 + Now it’s in the arctan form, with = √11:
3
1
3
3
=
arctan #
$=
arctan '
( + .
2 11 + 2√11
2√11
√11
√11
Example: Find
7
.
1 + 4 Solution: We prefer the quadratic term in the denominator to have a coefficient of 1. So factor a 4 from
the denominator, and move it (and the 7 while we’re at it) to the front:
Here, = , so that = :
%
7
7
1
= .
1 + 4
4 1⁄4 + 7
1
7 1
7
= #
$ arctan #
$ = arctan2 + .
4 1⁄4 + 4 1⁄2
1⁄2
2
The last step featured some basic simplification of fractions.
Be careful! Some forms look like the arctan form, but they are not:
1
= ln1 + + .
1+
2
Later, you will see some forms like:
.
1 + This one is solved by long dividing the denominator into the numerator:
1
= #1 −
$ .
1+
1 + The arctan then is used on the second part.
You may see a form such as
+ 2 + 4
.
& + 4
We use partial fractions:
+ 2 + 4
1
2
=
#
+
$ .
& + 4
+ 4
The arctan form is applied to the second term. You will see these forms later in this chapter.
Scott Surgent © 2014 surgent@asu.edu