6. Integration
6.1: Antiderivatives
Example 1. What is the revenue function R(x) if the marginal revenue is f (x) = R0 (x) = 2x − 1?
Generalization: Given a function f (x), want to find a function F (x) such that F 0 (x) = f (x). We call
F (x) an antiderivative of f (x).
DEFINITION Antiderivative
If F 0 (x) = f (x), then F (x) is called an antiderivative of f (x).
Theorem 1
On (a, b), if F 0 (x) = G0 (x), then
F (x) = G(x) + C,
where C is any constant.
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Example 2. Find all antiderivatives of 2x.
DEFINITION The Indefinite Integral
The collection of all antiderivatives of a function f (x) is called the indefinite integral and is denoted
by
Z
f (x) dx.
Given a function F (x) for which F 0 (x) = f (x), then
Z
f (x) dx = F (x) + C,
where C is an arbitrary constant and called the constant of integration.
R
We call f (x) the integrand. We refer to the symbol as the integral sign, which looks like an S
and stands for sum. How does an integral connect with a certain sum? We will see later.
Example 3. Find
Z
2x dx.
2
Rules of Integration
Power Rule
If n 6= −1, then
Z
1
xn+1 + C.
n+1
xn dx =
Proof.
Example 4. Using the power rule to find the following indefinite integrals. Check the answers.
a)
Z
x9 dx.
b)
Z
1
√ dx.
x3
3
Constant Times Function Rule
For any constant c,
Z
Z
cf (x) dx = c
f (x) dx.
Example 5. Using the Constant Times Function Rule to find the following indefinite integrals. Check
the answers.
a)
Z
b)
3x9 dx.
Z √ 3
s
ds.
ln3
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Sum or Difference Rule
Z
Z
(f (x) ± g(x)) dx =
Z
f (x) dx ±
g(x) dx.
Example 6. Using the Sum and Difference Rule to find the following indefinite integrals. Check the
answers.
a)
Z √
( t + t2 ) dt.
b)
Z r3
− r5
π
5
dr.
Indefinite Integral of Exponenial Function
Z
ex dx = ex + C.
Example 7. Find the following indefinite integral. Check the answer.
Z Indefinite Integral of x−1 =
eu
+ ue
4
du.
1
x
Z
1
dx = ln|x| + C.
x
Example 8. Find the following integral. Check the answer.
Z
8
− dw.
w
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Summary
Integration Rules
Z
Z
cf (x) dx = c f (x) dx
Z
Z
Z
(f (x) ± g(x)) dx = f (x) dx ± g(x) dx
Z
1
xn+1 + C, n 6= −1
xn dx =
n+1
Z
ex dx = ex + C
Z
1
dx = ln|x| + C
x
Appplications
Example 9. Given the marginal average cost 2 −
100
, find the average cost C(x).
x2
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Example 10. Given the marginal cost 5ex + 20x − 50 and the fixed cost 2000, find the cost function
C(x).
Example 11. Find f (x) using the following information.
f 0(x) = 12x2 − 8x + 2, f (4) = 208.
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