4.9 Antiderivatives
When we study applied sciences, we often find a function F whose
derivative is a known function f .
Definition
A function F is called an antiderivative of f on the interval I if
F 0 (x) = f (x) for all x ∈ I .
In general, it is not easy to find antiderivatives.
In this section we start with simple functions f .
Theorem
If F is an antiderivative of f on the interval I , then the most
general antiderivative of f on I is F (x) + C , where C is an arbitrary
constant.
From the previous theorem, we can see that the most general
antiderivative of f is a family of the function F .
Recall Differentiation Rules.
1
Derivative of a constant function:
d
(c) = 0 or (c)0 = 0
dx
2
The power Rule:
d n
(x ) = nx n−1 or (x n )0 = nx n−1
dx
3
The Contant Rule: for any constant c
d
d
(c f (x)) = c f (x) or (c f (x))0 = c f 0 (x)
dx
dx
4
The Sum(Difference) Rule:
d
d
d
(f (x) ± g (x)) =
f (x)± g (x), (f (x) ± g (x))0 = f 0 (x)±g 0 (x)
dx
dx
dx
We can use the Power Rule to obtain an antiderivative of x n :
d x n+1
(n + 1)x n
=
= x n,
dx n + 1
n+1
where n 6= −1. Thus the general antiderivative of f (x) = x n is
F (x) =
x n+1
+C.
n+1
For n = −1
F (x) = (ln |x|) + C , since
1
d
(ln |x|) =
dx
x
Every differentiation formlula gives rise to an
antidifferentiation formula, when you read it from right to left.
1
For the exponential function f (x) = e x
F (x) = e x
2
since (e x )0 = e x
For trigonometric functions
1. f (x) = sin x
F (x) = − cos x + C
since (− cos x)0 = sin x
2. f (x) = cos x
F (x) = sin x + C
since (sin x)0 = cos x
3. f (x) = sec2 x
F (x) = tan x + C
since (tan x)0 = sec2 x
Note that you can find antidifferentiation formulas in your
textbook.
Example1
Find the most general antiderivative of the function.
Check your answer by differentiation.
1.
f (x) = x − 1
2.
f (x) = 3x 2 − x + 2
3.
f (x) = 3x 1/2 − 5x 3/5
4.
√
√
f (x) = 4 x +8 x
5.
f (x) = (x − 1)(3x + 2)
6.
√
t5 + 3 t
f (t) =
t2
Example2
Find the most general antiderivative of the function.
Check your answer by differentiation.
1.
x 6 − x 3 + 2x
f (x) =
x4
2.
g (x) = 2 cos x + 3 sin x
Example3
1. Find the most general antiderivative of the function f (4) (t) = e t
2. Find f .
√
(1) f 0 (x) = x(6 + 5x), f (1) = 10
(2) f 00 (θ ) = 2 sin θ + cos θ , f (0) = 4, f 0 (0) = 6
(3) f 0 (x) = x 2 + sec2 x, f (π/4) = π 2 /16