Warm Up
Give an example of a function whose
derivative is:
1.
2.
3.
4.
5.
dy/dx = 6x2 + 1
dy/dx = cos x
dy/dx = 1/x
dy/dx = ex
dy/dx = 0
Many operations in mathematics have
inverses. The reverse operation of
finding a derivative is called the
antiderivative. F is an antiderivative
of f if F ’(x) = f (x).
CHAPTER FOUR:
INTEGRATION
•
•
•
•
•
•
•
Sect. 4.1 Antiderivatives
Sect. 4.2 Area
Sect. 4.3 Riemann Sums/Definite Integrals
Sect. 4.4 FTC and Average Value
Sect. 4.5 Integration by Substitution
Sect. 4.6 Numerical Integration and Trapezoidal Rule
Particle Motion
Sect. 4.1 Antiderivatives and
Indefinite Integration
• Write the general solution of a differential
equation.
• Use indefinite integral notation for
antiderivatives.
• Use basic integration rules to find
antiderivatives.
• Find a particular solution of a differential
equation.
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.
DIFFERENTIAL
EQUATIONS
• A differential equation
is an equation that
contains a derivative.
• By antidifferentiating,
we can solve this
equation for y.
x 2
3
x 3
dy
2
 3x
dx
yx
3
3
x  100000000
3
Indefinite Integrals
The family of all functions that are antiderivatives
of f (x) is called the indefinite integral.
Integrand

Constant of
Integration
f  x  dx  F  x   C
Integral
Sign
Variable of Integration
Read: The antiderivative of f with respect to x or
the indefinite integral of f with respect to x is
equal to…..
Some General Rules
They are just the derivative rules in
reverse.
Differentiation Formula
d
C   0
dx
d
 kx   k
dx
d
k f  x  k f  x
dx
Integration Formula
 0 dx  C
 k dx  kx  C
 k f  x  dx  k  f  x  dx
Some More General Rules
Differentiation Formula
Integration Formula
d
f  x   g  x   f  x   g x 

dx
  f  x   g  x  dx   f  x  dx   g  x  dx
Sum / Difference Rule for Integrals
 
d n
x  nx n1
dx
n 1
x
 x dx  n  1  C
n
Power Rule for Integrals
Back to Warm Up
Integrate the following:
 6 x
2

 1 dx  2 x 3  1x  c
 cos x dx
 sin x  c
1
  x dx  ln x  c
x
x
e
dx

e
c

 0dx
c
You Try…
Integrate the following.
1. 6a da  6a  C

2.  dg
3.  3m
4.
 g C
3m5 5m3 m2



C
4
 5m 2  m dm
5
3
2
2
t
1
3


C
1

t
dt


C
 
2
2
 3 dt
2t
 t



 1 4 3   (z
5.  
 z dz 
 z

1 /2
12
7 4
4z
z
z
12
3/ 4

2
z


C
 z )dz 
12 7 4
7
7 4
C
OTHER INTEGRALS THAT
SHOULD BE KNOWN
 cos ax
C
a
sin ax
cos
axdx

C

a
 sin axdx 
2
sec
 xdx  tan x  C
 sec x tan xdx  sec x  C
 csc x cot xdx   csc x  C
 csc xdx   cot x  C
2
kx
e
kx
e
 dx  k  C
k x
a
kx
a
 dx  k  ln a   C
dx
 x  ln x  C
dx
 1  x2  arcsin x  C
dx
 1  x2  arctan x  C
dx
 x x2  1  arcsec x  C
Rewriting before integrating
 x 4  8 x3 
1. 
 dx 
2
 x

 x
2

 8 x dx 
3
x
 4 x2  C
3
2.


2
 
x
x

3
dx

4
2
x
x
3
x  3 x dx   3  c
4
2

You Try…
Find the antiderivative.
x 1
x
1
1.
dx  

dx
x
x
x
32
2
x
 2 x1 2  C
  x1 2  x 1 2 dx 
3
2. ( x  2)( x  3)dx 
2
x
3
 3x  2 x  6  dx
2
x4

 x3  x2  6 x  C
4
3.
 2 x  5 3x  2 dx  12 x5/2 22 x3/2  20 x1/2  C
x
5
3
White Board Practice
1.  2dx  2 x  C
 x2

2. 12( x  1)dx  12  x   c
 2


1
1
3.  3 dx   2
x
2x
 u 7

2
4.   3e   2u  6 du
u


5.

2 3
 3e  7 ln u  u  6u  C
3
u
7
3
3 3
x  x  4 dx  x  3x 4 / 3  C
7
4
3
6 
 3
6.   8 x 
dx  6 x  12 x  C
x

Day 2
Sect. 4.1 Integration
More Integration
1
cos(5 x)  5 sin(5 x)  C
1
sin(6 s  9)ds   - cos(6s  9)  C
6

2. 
2t
3.  16e dt
1.
 8e 2t  C
5x
8
C
4. 85 x dx 
5ln 8
2
5. 5 x  9  dx  1  1 5 x  93  1 5 x  93
3 5
15
1 
1 1

2
2

xdx

2
x
dx

dx
6.  x  2 x  dx 


3 x
3x 




x2
x3 1

 2  ln x  C
2
3 3
You Try…
9t
e
1.  e dt 
c
9
9t
5
3
 x4 x2 
x
x
2.    dx 
 c
2
20 6
 4


3.  (6 x 3  4 x) 2 dx   36 x 6  48 x 4  16 x 2 dx
36 7 48 4 16 2
5 k
2
1

x 
x  x c
4.  2 5 k dk 
  c
7
5
3
ln 2 5
 2 7x
2
e7 x
5. 
 e dx   ln x 
c
3x
3
7
Initial Conditions and
Particular Solutions
When we find antiderivatives, we are creating
a family of curves for each possible value of c,
called the general solution.
 3x
C=0
2
y  x3  x
C=1
y  x3  x  1
C=3
y  x  x 3
3

 1 dx  x 3  x  C
When we find the
value of C,
given a point on y
(called an initial
condition), we
are able to find a
particular solution.
Example 1
Find the equation of the curve
that passes through (2,6) if its
slope is given by dy/dx = 3x2
at any point x.
 3x dx 
2
 x3 
3   C  x 3  C
 3
y  x C
3
6  23  C
C  2
Therefore, y = x3 - 2
Ex 2: Find the general solution of the equation
1
F’(x) = 2 . Then find the particular solution given
x
the point F(1) = 0.
1
1
2
F ( x )   2 dx   x dx    C
x
x
Now plug in (1,0) and solve for C.
0 = -1 + C
1
F ( x)    1
x
C=1
You Try…
Find the function f (x) for which
f  x   x 2  x and f (1) = 3.
x3 2 x3 2
f  x  
C
3
3
13 2 13 2
1 2 1
3  f 1  
C  
 C  1 C
3
3
3 3
So, 3 = 1 + C and therefore, C = 2.
Therefore, our function is
x 3 2 x3 2
f  x  
2
3
3
You Try…
Given: f x   x 2 , f 0  6, f (0)  3
Find f(x)
Integrate:
f  x   x 2
x3
f  x    c
3
Substitute x=0 and 6 for f’(0)
Thus c = 6
x3
f x    6
3
Integrate again:
1 x4
x4
f x     6 x  c 
 6x  c
3 4
12
Substitute x = 0 and f(0)=3
Answer:
03
6 c
3
4
x
f x  
 6x  3
12
04
3   60  c
12
c= 3
Closure
• Explain how to integrate
x  2x  6
dx
 x
3
• Explain how to check your
answer the problem above.
SUMMARY for Integration
• A differential equation is an equation that
contains a derivative.
• You can solve them analytically using
integration.
• The integral of a derivative is called the
antiderivative. Don’t forget to include an
arbitrary constant.
• There are infinite solutions to a differential
equation, but with an initial value you can
solve for an arbitrary constant to help you
calculate the particular solution.
• You SHOULD memorize the table of integrals
on pgs. 244 & 391!! In addition, memorizing
derivatives are very, very important!