1. 4.9 Definitions and Properties
(1) A function F is called an antiderivative of f on an interval I if F 0 (x) = f (x)
for all x ∈ I.
(2) Theorem: If F is an antiderivative of f on an interval I and C is any constant,
then F (x) + C also defines an antiderivative of f on I.
(3) If F is an antiderivative of f , then we describe the antiderivative of a function
in the most general terms by using the notation F (x) + C to represent all
possible antiderivatives of f .
Example 1.1. Find an antiderivative of f (x) = x4 .
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4.9 AntiDerivatives
2. Rules for the most general antiderivative of f – Memorize
f
The general antiderivative of f
f (x) = k where k is a constant
f (x) = kg(x) where k is a constant
f (x) = g(x) + h(x)
f (x) = xn for n 6= −1
f (x) = x−1
f (x) = ex
f (x) = ax where a > 0 and a 6= 1
f (x) = sin x
f (x) = cos x
f (x) = sec2 x
f (x) = sec x tan x
f (x) =
1
1 + x2
2
4.9 AntiDerivatives
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3. Examples
Example 3.1. Find the most general antiderivative for f (x) =
Example 3.2. Find the most general antiderivative for f (x) =
√
3
x.
1
.
3 cos2 x
Example 3.3. Find the most general antiderivative for f (x) = xπ + π x .
Example 3.4. Find the most general antiderivative for f (x) = 2x3 + 4x2 + 3x + 1.
Example 3.5. Find the most general antiderivative for f (x) = (2x + 1)(x − 3).
4.9 AntiDerivatives
4
√
x4 + x − 4 3 x
Example 3.6. Find the most general antiderivative for f (x) =
.
x2
Example 3.7. Find the antiderivative F of f (x) =
F (1) = π4 .
x2
8
that satisfies the condition
+1
√
Example 3.8. Find f given f 00 (x) = 3/ x, f (4) = 20, and f 0 (4) = 7.
4.9 AntiDerivatives
5
Example 3.9. A particle is moving with a(t) = cos t+sin t, s(0) = 2, and s(π) = −1.
Find the position when t = π/2.
Example 3.10. Which of the following curves could be an antiderivative of graph of
the function given by curve C.
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A 1
B K2
0
K1
K1
C K2
1
x
2