W15 MAT21B LECTURE 2: ANTIDERIVATIVES, BEGINNING INTEGRATION
GEORGE MOSSESSIAN
By now, you know how to take derivatives of functions, and it stands to reason that if you can take a
derivative, you should be able to undo the taking of a derivative as well. This is called
taking an antideriva
tive
. Notice the word
an
, as opposed to
the
, which suggests that there may be more than one. Let’s see why
this is true.
Consider
y
= 2
x
. Is there a function
F
(
x
) such that
F
0
(
x
) = 2
x
? Of course, you say,
F
(
x
) =
x
2
. Except
that
x
2
+ 1 also works, as does
x
2
+
C
for any real number
C
. Thus, there are actually an
infinite number
of antiderivatives of 2
x
, but they all
1
look like
x
2
+
C
.
Another example:
y
= cos
x
has antiderivative sin
x
+
C
,
y
= sin
x
has antiderivative

cos
x
+
C
, and
y
= 2
x
+ sin
x
has antiderivative
x
2

cos
x
+
C
.
In general, the antiderivative of
x
n
is
1
n
+ 1
x
n
+1
+
C
. You can check this by taking the derivative of that
result:
1
n
+ 1
x
n
+1
+
C
0
=
1
n
+ 1
·
(
n
+ 1)
x
n
+ 0 =
x
n
.
When there are constants involved, you have to “unchain” them, that is, make sure they get cancelled
out in the chain rule that follows from taking the derivative, like so:
(sin
kx
)
0
=

1
k
cos(
kx
)
,
so that when you take the derivative of

1
k
cos
kx
, the
k
which results from chain rule on
kx
cancels out
with the
1
k
.
This leads to a sort of general philosophy of finding antiderivatives: first, make an educated guess which
may or may not have the correct constant coefficients, and then adjust for those. For example: if I want to