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Lecture 31: Sec. 7.1
The Antiderivative
ex. Suppose that the velocity (in cm/sec) of an
object moving along a straight line after starting from
rest is given by v(t) = 3t2 + 2t. Can we find the
position function s(t)?
Def. A function F is an antiderivative of function f on an interval I if for each x in that interval,
The operation of finding the antiderivative is called
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ex. Show that F (x) = ln(x2 +3) is an antiderivative
2x
.
of f (x) = 2
x +3
ex. Find an antiderivative of f (x) = x3 + 4. How
many antiderivatives can there be?
Theorem Let F be an antiderivative of a function
f . If G is another antiderivative of f
(that is, G0(x) = F 0(x) =
), then
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ex. Find all antiderivatives of f (x) = −2x.
6
-
ex. Find the unique antiderivative F (x) satisfying
the condition that F (1) = −2.
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The Indefinite Integral
Def. Let F (x) be an antiderivative of f (x), so that
F 0(x) = f (x).
Then we represent the family of functions which are
antiderivatives of f by the indefinite integral
R
is
f (x) is
C is
Note the use of the differential dx.
Z
ex.
(x3 + 4) dx =
How to find antiderivatives?
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Basic Integration Rules
Z
1. For any real number k, k dx =
ex. Evaluate:
Z
(a) π dx
Z
(b)
3 dt
ex. Evaluate:
Z
(a) x dx
Z
(b)
x2 dx
2. The Power Rule
Z
If n 6= −1, then xn dx =
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Z
ex.
x54 dx =
3. The Constant Multiple Rule
Z
If k is a constant, k f (x) dx =
NOTE:
4. The Sum or Difference Rule
Z
[f (x) ± g(x)] dx =
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ex. Evaluate each indefinite integral:
Z
(a) (6x2 − x5 + 5) dx
Z
(b)
Z
(c)
9
dx
2x4
√
2x + 3 x + 1
√
dx
x
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5. Integrals of Exponential Functions
Z
(a) ex dx =
(b) For
Z any constant k,
ekx dx =
Z
(c)
ax dx =
6. Integral of f (x) =
Z
1
dx =
x
NOTE:
Z
1
x
x−1 dx =
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Z ex. Evaluate:
1
1
3−
1+
dx
x
x
Particular Solutions
ex. Find the unique function f (x) so that
f 0(x) = e0.2x − 3 and f (0) = 2.
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Applications of the Indefinite Integral
ex. A company has determined that the marginal
revenue for a certain product is R0(x) = 150 − 3x.
Find the revenue function if the revenue from the sale
of 20 items is $2400. What is the demand function
p(x) for the product?
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ex. Suppose that a snail is moving along a straight
path with acceleration a(t) = 3t − 4 cm/sec2. If its
initial velocity is 6 cm/sec find its distance from the
starting point after 2 seconds.
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Additional Example
ex. The slope of the curve y = f (x) at any point
√
( x − 2)2
(x, f (x)) is given by
. Find the function
x
f if its graph passes through the point (1, −3).
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Now you try it! Problems are on pages 13 and 14
1. Find the following antiderivatives:
Z
Z 6
(a) x−1/3 (6 − x) dx
(b)
e−0.4x − 4x +
dx
x
2. A company estimates that the marginal cost (in dollars per item)
of producing x items is 1.6 − 0.08x. If the cost of producing ten
items is $480, find the cost of producing 20 items. What are the
fixed costs of the product?
3. Sales of a new product are growing at the rate S 0 (t) = 40t1/4 − 12
where S represents the monthly sales in hundreds t months after
the product is introduced. If 800 units were sold in the first month
(S(1) = 8), find the expected sales after 16 months.
4. Find the function g(x) if the slope of the tangent line to g(x) at
√
( x + 1)2
√
and g(4) = 2.
any x is given by
x
5. A small rocket shot from the ground experiences an acceleration of
5 − 2t ft/sec2 .
(a) Find a formula for the height h of the rocket, in feet, after t
seconds, if its initial velocity is 50 ft/sec.
(b) What is the maximum height reached by the rocket? When is
the rocket at this point?
6. A car is traveling at 66 ft/sec (45 mi/hr) when the brakes are fully
applied, producing a constant deceleration of 22 ft/sec2 . What
distance is traveled by the car before it comes to a stop? Hint: Let
t = 0 when the brakes are first applied. What are v(0) and s(0)?
Note that a(t) = −22. What is the velocity when the car comes to
a stop?
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7. The graph of y = f (x) is sketched below. Make a rough sketch of
the antiderivative F (x) of f so that F (0) = −2.