ANTIDERIVATIVES v.03
d f HxL
If d x = gHxL, or what is the same f ¢ HxL = gHxL on an open interval I, f HxLis said to be an
antiderivative of gHxL.
These are examples of antiderivatives of some functions. It can be double checked that they are
indeed antiderivatives by calculating the derivative of the resulting function.
Function
Some Antiderivatives
gHxL = x
f HxL =
1
2
x2
gHxL = x , x > 0
1
f HxL = LnHxL
gHxL = 2x
f HxL =
hHxL =
1
2
x2 + 3
rHxL =
hHxL = LnHxL + 10
1
LnH2L
2x
hHxL =
1
LnH2L
1
2
x2 - 5.7
rHxL = LnHxL - 3
2x + 3
rHxL =
1
LnH2L
2x - 2
As a consequence of the Mean Value Theorem, functions with the same derivative differ only
by a constant. It is, if f HxL is an antiderivative for gHxL, then any other antiderivative for gHxL is
of the form f HxL + C , C a constant.
These are some antiderivatives of f HxL = x
Antiderivative of y=x
Antiderivative of y=x
12
10
8
6
4
2
0
-2
12
10
8
6
4
2
0
-2
-3
-2
-1
0
1
2
3
-3
Antiderivative of y=x
-2
-1
0
1
2
3
Antiderivative of y=x
12
10
8
6
4
2
0
-2
12
10
8
6
4
2
0
-2
-3
-2
-1
0
1
2
3
-3
-2
-1
0
1
2
3
2
30IntegralAntiderivatives.nb
The family of functions gHxL =
gHxL =
1
2
1
2
x2 + C , C constant, are antiderivatives for f HxL = x;
x2 + C is called the general antiderivative of f HxL = x and is denoted as
Ù x âx =
1
2
x2 + C
Notice that each value of C will yield a different antiderivative. Since these antiderivatives are
obtained by vertical translations, they do not intersect. It means that if we choose a point in the
plane, and there is an antiderivative passing through that point, it has to be unique.
EXAMPLE 1
Find the antiderivative of f HxL = x passing through the point (1, -2).
Since the general antiderivative is of the form gHxL =
that (1, -2) is on that function. Need to solve -2 =
are looking for is gHxL =
1
2
x2 -
1
2
1
2
x2 + C, we need to find the value of C such
5
H1L2 + C, C = - 2 . Hence, the antiderivative we
5
2
EXERCISE 1
a. Find the antiderivative of f HxL = x + 2 passing through the point (2, -2).
b. Find the antiderivative of gHxL = 3 - x2 passing through the point (0,4)
PROPERTIES OF ANTIDERIVATIVES
Antiderivatives have the same basic properties as derivatives:
1. Ù c f HxL â x = c Ù f HxL â x, where c is a constant
2. Ù H f HxL + gHxLL â x = Ù f HxL â x+Ù gHxL â x
BASIC ANTIDERIVATIVES
This is the list of most common antiderivatives we will
encounter:
1. Ù â x = x + C
2. Ù xn â x =
3. Ù
1
x
xn+1
n+1
+ C, n ¹ -1
â x = Ln x +C
30IntegralAntiderivatives.nb
This is the list of most common antiderivatives we will
encounter:
1. Ù â x = x + C
2. Ù xn â x =
3. Ù
1
x
xn+1
n+1
+ C, n ¹ -1
â x = Ln x +C
4 Ù ãx â x = ãx + C
5. Ù ax â x =
1
LnHaL
ax + C, a > 0
6. Ù cosHxL â x = sinHxL + C
7. Ù sinHxL â x = -cosHxL + C
8. Ù sec2 HxL â x = tanHxL + C
9. Ù secHxL tanHxL â x = secHxL + C
10. Ù
1
1+x2
11. à
â x = arctanHxL + C
1
â x = arcsinHxL + C
1-x2
Find following antiderivative :
1. Ù 3 â x
2. Ù Hx - 0.5L â x
1
3. Ù Jx2 + x - x + 2N â x
4. Ù H3 - sinHtLL â t
5. Ù Isec2 HxL - cosHxLM â x
6. Ù Iex + 2x - x2 + xã+3 - ΠM â x
7. à J2 x7.5 +
2
x
- 2N â x
3