Indefinite Integrals
Consider a continuous function f . If F is an antiderivative for f on [a, b], then

(1)
b
a
f  x  dx   F  x  a
b
If C is a constant, then
 F  x   C  a   F  b   C    F  a   C   F  b   F  a    F  x  a
b
b
Thus we can replace (1) by writing
 f  x  dx   F  x   C 
b
b
a
a
.
If we have no particular interest in the interval [a, b] but wish instead to emphasize that
F is an antiderivative for f , which on open intervals simply means that F´ = f , then
we omit the a and the b and simply write
 f  x  dx  F  x   C
Antiderivatives expressed in this manner are called indefinite integrals. The constant
C is called the constant of integration; it is an arbitrary constant and we can assign
to it any value we choose. Each value of C gives a particular antiderivative, and each
antiderivative is obtained from a particular value of C.
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Salas, Hille, Etgen Calculus: One and Several Variables
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Indefinite Integrals
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Indefinite Integrals
The linearity properties of definite integrals also hold for indefinite integrals.
Example 1
3/ 2
2
Calculate  5 x  2csc x  dx
Solution
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Indefinite Integrals
Example 2 Find f given that
f x   x 3  2 and f 0  1.
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Indefinite Integrals
Example 3 Find f given that
f " x  6 x  2,
f ' 1  5,
and f 1  3.
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Indefinite Integrals
Application to Motion
Example 4
An object moves along a coordinate line with velocity
v(t) = 2 − 3t + t2 units per second.
Its initial position (position at time t = 0) is 2 units to the right of the origin. Find
the position of the object 4 seconds later.
Solution
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Indefinite Integrals
Example 5 An object moves along the x-axis with acceleration a(t) = 2t - 2 units per
second per second. Its initial position (position at time t = 0) is 5 units to the right of
the origin. One second later the object is moving left at the rate of 4 units per
second.
(a) Find the position of the object at time t= 4 seconds
(b) How far does the object travel during these 4 seconds?
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Indefinite Integrals
Example 6 Find the equation of motion for an object that moves along a straight
line with constant acceleration a from an initial position x0 with initial velocity v0.
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The u-Substitution
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The u-Substitution
Example 1 Calculate
2
4
(
x

1
)
dx.

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The u-Substitution
Example 2 Calculate
 sin
2
x cos x dx.
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The u-Substitution
Example 3 Calculate
 2x
2


sin x  1 dx.
3
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The u-Substitution
Example 4
Calculate
1
 3  5x 
2
dx
Solution
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The u-Substitution
Example 5 Calculate
2
3
x
4

x
dx.

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The u-Substitution
Example 6 Calculate


3
2
4
2
x
sec
x
 1 dx.

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The u-Substitution
Example 7. Calculate
3
sec
 x tan x dx.
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The u-Substitution
Example 8 Evaluate
 x
2
0
2


2
 1 x  3x  2 dx.
3
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The u-Substitution
Example 9 Evaluate
 x
2
0
2


2
 1 x  3x  2 dx.
3
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The u-Substitution
Example 10 Evaluate
1/ 2

0
cos  x sin  dx.
3
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The u-Substitution
Example 11 Calculate
 xx  3 dx.
5
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The u-Substitution
Example 12. Evaluate

0
3
x 5 x 2  1 dx.
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The u-Substitution
The Definite Integral
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