6A (This accompanies part of section 7.2 of the book): Antidifferentiation
Example 1: What functions have the following derivatives?
(a) 3𝑥 2
(c) 15
(b) 4𝑥 3
(d)
1
𝑥
(e) sec2x
(f)
1
√1−𝑥 2
Antidifferentiation: the process of finding a function 𝑓 whose known derivative is 𝑓 ′ (𝑥).
Definition: Indefinite Integral
The family of all antiderivatives of a function 𝑓(𝑥) is the indefinite integral of 𝒇 with respect to x and is
denoted:
∫ 𝑓(𝑥)𝑑𝑥
If 𝐹 is any function such that 𝐹 ′ (𝑥) = 𝑓(𝑥), then ∫ 𝑓(𝑥)𝑑𝑥 = 𝐹(𝑥) + 𝐶 where 𝐶 is an arbitrary constant
called the constant of integration.
** the symbol ∫ is called the integral sign; f is called the integrand, x is the variable of integration
Properties of Indefinite Integrals:
1.
∫ 𝑘𝑓(𝑥)𝑑𝑥 = 𝑘 ∫ 𝑓(𝑥)𝑑𝑥
2. ∫( 𝑓(𝑥) ± 𝑔(𝑥) )𝑑𝑥 = ∫ 𝑓(𝑥)𝑑𝑥 ± ∫ 𝑔(𝑥)𝑑𝑥
Example 2: Find the most general antiderivative of the function:
1
𝑓(𝑥) = 𝑥 + 𝑒 𝑥 − cos 𝑥
Power Formula:
∫ 𝑢𝑛 𝑑𝑢 =
𝑢𝑛+1
𝑛+1
+𝐶
Example 2: Find the most general antiderivative for the function. Check you answer by differentiating:
𝑔(𝑥) = 𝑥 5 − 4𝑥 −3 + √𝑥 + 20 – csc 𝑥 cot 𝑥
**The constant of integration 𝐶 can be found if certain initial conditions exist as shown in the next
example.
Example 3: Find the antiderivative 𝐹 that satisfies the given conditions:
𝑓(𝑥) = 3𝑥 − sin 𝑥 and 𝑓(0) = 3
Example 4: Find 𝑓.
𝑓 ′′ (𝑥) = 20𝑥 3 − 10
𝑓(1) = 1, 𝑓 ′ (1) = −5