Section 5.8: Antiderivatives
Definition: A function F is called an antiderivative of f on an interval if F 0 (x) = f (x) for
all x in that interval.
Example: Show that F (x) = x ln x − x is an antiderivative of f (x) = ln x on (0, ∞). Are
there any other antiderivatives of f ?
Theorem: If F is an antiderivative of f on an interval, then the most general antiderivative
of f is F (x) + C, where C is an arbitrary constant.
The following table lists some important antiderivatives.
Function
Antiderivative
Function
Antiderivative
k
kx + C
1
x
ln |x| + C
xn , n 6= −1
xn+1
+C
n+1
ex
ex + C
cos x
sin x + C
sec x tan x
sec x + C
sin x
− cos x + C
csc x cot x
− csc x + C
sec2 x
tan x + C
csc2 x
− cot x + C
1
+1
tan−1 x + C
√
1
1 − x2
sin−1 x + C
x2
Theorem: Suppose that F (x) and G(x) are antiderivatives of f (x) and g(x), respectively.
1. The function cF (x) is an antiderivative of cf (x), where c is any constant.
2. The function F (x) ± G(x) is an antiderivative of f (x) ± g(x).
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Example: Find the most general antiderivative of each function.
(a) f (x) = x3 − 2x2 + 5x − 6
(b) f (x) =
√
√
3
x2 − x3
(c) f (x) =
x2 + 2x + 3
x
(d) f (x) = ex + sin x − sec x tan x
2
(e) f (x) = ex/2 + e−x/2
(f) f (x) = sin
(g) f (x) =
x
4
+ cos(3x)
1
1 + 3x
(h) f (x) = sin2 x + cos2 x
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Initial-Value Problems
Example: Find f (x) if f 0 (x) = 12x2 − 24x + 1 and f (1) = −2.
Example: Find f (x) if f 00 (x) = 3ex + 5 sin x, f (0) = 1, and f 0 (0) = 2.
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Example: A stone is dropped from the upper observation deck of a tower 450 m above the
ground. Find the distance of the stone above ground level at time t.
Example: A car is traveling at 80 ft/s when the brakes are fully applied, producing a constant
deceleration of 40 ft/s2 . What is the distance covered before the car comes to a stop?
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Example: Suppose that the length (in cm) of a certain organism at age x is given by L(x),
which satisfies the differential equation
dL
= e−0.1x ,
dx
x ≥ 0.
Find L(x) if the limiting length L∞ is given by
L∞ = lim L(x) = 25.
x→∞
How large is a newborn organism?
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