5_2018_12_22!12_52_06_AM

1. Indefinite Integral

2. Introduction

A physicist who knows the velocity of a particle
might wish to know its position at a given time.
A biologist who knows the rate at which a
bacteria population is increasing might want
to deduce what the size of the population will be
at some future time.

3. Antiderivatives

In each case, the problem is to find a function F
whose derivative is a known function f.
If such a function F exists, it is called an
antiderivative of f.
Definition
A function F is called an antiderivative of f on
an interval I if F’(x) = f (x) for all x in I.

4. Find the integral. (Find the antiderivative.)

x
dx
=
?
n
1
n 1
x
n 1
C

5. Antiderivatives

Theorem
If F is an antiderivative of f on an interval I, then
the most general antiderivative of f on I is
F(x) + C
where C is an arbitrary constant.
Going back to the function f (x) = x2, we see that
the general antiderivative of f is ⅓ x3 + C.

6. Notation for Antiderivatives

The symbol f ( x )dx is traditionally used to
represent the most general an antiderivative of f
on an open interval and is called the indefinite
integral of f .
Thus, F ( x) f ( x)dx means F’(x) = f (x)
3
x
2
x
dx 3 C
because the derivative of
x3
2
is
x
C
3

7.

Constant of Integration
Every antiderivative F of f must be of the form
F(x) = G(x) + C, where C is a constant.
Example:
2
6x
dx
3x
C
Represents every possible
antiderivative of 6x.

8.

Power Rule for the Indefinite
Integral
n 1
x
x dx
C if n 1
n 1
n
Example:
4
x
3
x dx
C
4

9.

Power Rule for the Indefinite
Integral
1
x dx dx ln x C
x
1
Indefinite Integral of ex and bx
e dx e C
x
x
x
b
b dx
C
ln b
x

10. Sum and Difference Rules

f
g
dx
fdx
gdx
Example:
3
2
x
x
x x dx x dx xdx C
3
2
2
2

11.

Constant Multiple Rule
kf ( x)dx k f ( x)dx (k constant)
Example:
4
4
x
x
2 x dx 2 x dx 2 C
C
4
2
3
3

12. Integration by Substitution

Method of integration related to chain rule. If u
is a function of x, then we can use the formula
f
f dx
du
du / dx

13.

Integration by Substitution
Example: Consider the integral:
3x x 5 dx
2
9
3
pick u x +5, then du 3x dx
3
9
u du
Sub to get
2
x 5
u
C
10
10
10
Integrate
3
C
10
Back Substitute

14.

Example: Evaluate
x 5x 7dx
2
Let u 5x 7, du 10x dx
Pick u,
compute du
1
1 12
2
10 10x 5x 7 dx 10 u du
Sub in
2
1 u
C
10 3/ 2
3/ 2
5x 7
Integrate
3/ 2
2
15
C
Sub in

15.

Example: Evaluate
dx
x ln x
3
Let u ln x then xdu dx
dx
x ln x u du
3
3
u 2
C
2
ln x
2
2
C

16.

Examples on
your own:

17.

Find the integral of each:
1.) 8 dx
2.) (2x 6) dx
2x 2
F(x)
6x c
2
F(x) 8x c
dx
(6x
12x
8)
3.)
2
(9x
12x 9) dx
4.)
2
6x 12x
F(x)
8x c
3
2
3
F(x) x 2 6x c
2
F(x) 2x 3 6x 2 8x c
9x 3 12x 2
F(x)
9x c
3
2
F(x) 3x 3 6x 2 9x c

18.

Find the integral of each:
5.) (x 2)10 dx
6.) 2(2x 3) 4 dx
u du
u du
u x 2
u 2x 3
4
10
11
du 1 dx F(x) u C
11
11
(x 2)
F(x)
C
11
6
(5
x)
dx
7.)
u 5 x
du 1 dx
u du
6
1 u 6 du
u7
F(x) C
7
5
u
du 2 dx
F(x) C
5
(2x 3) 5
F(x)
C
5
(3x 1) 4 dx
8.)
u 3x 1
du 3 dx
(5 x) 7
F(x)
C
7
1
1 4
4
u
du
u
du
3
3
1 u5
F(x)
C
3 5
(3x 1) 5
F(x)
C
15

19.

Find the integral of each:
9.) 1 dx
10.)
3
dx
x 4
x 2
u x 2
u x 4
1
u du
du 1 dx
du 1 dx
F(x) ln u C
11.)
3x
u x 5
3
du 3x 2 dx
2
x 5 dx
3
u du u
3
F(x) 3ln u C
F(x) 3ln x 4 C
F(x) ln x 2 C
1
3 du
u
1
2
du
u 2
F(x)
C
3
2
3
2 3
F(x) x 5 2
C
3
12.) x x 5 dx
3
u x 5
du 2x dx
2
2
1
u 3
2 du
4
1 u 3
F(x)
C
4
3 2
2 4
3
3
F(x)
x
5
C
8

20.

Find the integral of each:
4
10
13.)
14.) 3 dx
dx
x
x2
10x
2
10 x
2
4x
du
du
1
x
F(x) 10
C
1
10
F(x)
C
x
3
du
4 x 3 du
x 2
F(x) 4
C
2
2
F(x) 2 C
x