Aim: How do we integrate the natural
logarithmic function?
Do Now:
Find the derivative
y  ln  ln x
2

Aim: Integrating Natural Log Function
Course: Calculus
Log Rule for Integration
Rules of
Differentiation
Rules of
Integration
d
1
ln x   , x  0

dx
x
d
1 du u '
ln u 
 ,u0

dx
u dx u
Let u be a differentiable
function of x
1
 x dx  ln x  C
1
 u du  ln u  C
Enables integration of rational functions
Aim: Integrating Natural Log Function
Course: Calculus
Model Problems
2
2
1
 x dx  2 xdx  2ln x  C  ln x  C
 ln x 2  C
1
 4 x  1 dx u = 4x – 1 u’ = 4
1  1 
Multiple &
 
4
dx

Divide by 4
4  4x  1 
1 1
    du
Substitute u
4  u
1
 ln u  C
4
1
 ln 4 x  1  C
4
Aim: Integrating Natural Log Function
Log rule
Back substitute
Course: Calculus
Model Problem
1
u'
 u du  ln u  C   u dx  ln u  C
Alternate form
of Log Rule
du = u’dx
Look for quotients in which numerator is
the derivative of denominator.
Find the area of the region bounded
x
by the graph y  2
, the x -axis, and the
x 1
x
line x  3.
h x  =
x +1
1.2
1

3
0
x
dx
2
x 1
2
0.8
0.6
0.4
0.2
1
Aim: Integrating Natural Log Function
-0.2
2
Course: Calculus
3
Model Problem
u'
 u dx  ln u  C
u’ = 2x
u = x2 + 1

3
0
Look for quotients in which
numerator is derivative of
denominator.
x
1 3 2x
dx   2
dx
2
x 1
2 0 x 1


3
1
2
  ln x  1 
0
2
1
  ln10  ln1
2
1
 ln10
2
 1.151
Aim: Integrating Natural Log Function

u'
dx  ln u  C
u
ln 1  0
Course: Calculus
Model Problems
u'
 u dx  ln u  C
3 x2  1
 x 3  x dx
sec 2 x
 tan x dx

x1
dx
2
x  2x
Look for quotients in which
numerator is derivative of
denominator.
u = x3 + x
u’ = 3x2 + 1
u = tanx
u’ =
sec2x
u = x2 + 2x
u’ = 2x + 2
 ln x 3  x  C
 ln tan x  C
1 2x  2
  2
dx
2 x  2x
1
 ln x 2  2 x  C
2
Aim: Integrating Natural Log Function
Course: Calculus
Model Problems
Look for quotients in which
u'
is a degree higher or
 u dx  ln u  C numerator
equal to denominator
x2  x  1
long
division
dx
 x2  1
x2  x  1
x
2
2
 x 1 x  x 1  1 2
2
x 1
x 1
x 
x

  1  x 2  1 dx   1dx   x 2  1 dx
1
2x
u = x2 + 1
 x  C1   2 dx
u’ = 2x
2 x 1
1
 x  ln x 2  1  C
2
Aim: Integrating Natural Log Function
Course: Calculus
Aim: How do we integrate the natural
logarithmic function?
Do Now:
Evaluate:

x2  4
dx
x
Aim: Integrating Natural Log Function
Course: Calculus
Model Problem – Change of Variables
u'
 u dx  ln u  C
2x
  x  1 dx
2
Look for quotients in which
numerator is derivative of
denominator.
u=x+1  x=u–1
du
 1  du  dx
dx
2  u  1

du
2
u
Substitute u
1 
 u
 2  2  2 du
u 
u
Rewrite 2
fractions
u
1
Rewrite 2
 2 2 du  2 2 du
Integrals
u
u
Aim: Integrating Natural Log Function
Course: Calculus
Model Problem – Change of Variables
u'
 u dx  ln u  C
Look for quotients in which
numerator is derivative of
denominator.
2x
u
1
dx
  x  12  2 u2 du  2 u2 du
du
 2
 2 u2du
u
 u 1 
 2ln u  2 
C

 1 
Rewrite 2
Integrals
Integrate
2
 2ln u   C
Simplify
u
2
Back 2ln x  1 
C
substitute
x1
Aim: Integrating Natural Log Function
Course: Calculus
Guidelines for Integration
1. Memorize a basic list of integration
formulas. (20)
2. Find an integration formula that resembles
all or part of the integrand, and, by trial and
error, find a choice of u that will make the
integrand conform to the formula.
3. If you cannot find a u-substitution that
works, try altering the integrand. You
might try a trig identity, multiplication and
division by the same quantity, or addition
and subtraction of the same quantity. Be
creative.
4. If you have access to computer software that
will find antiderivatives symbolically, use it.
Aim: Integrating Natural Log Function
Course: Calculus
Model Problem
dy
1
Solve the differential equation

dx x ln x
1
Will log rule apply? u
y
dx
x ln x
What does u equal? u
n 1
n
u'
u = x  dx u = x lnx
u
y
u'
u'
 u dx u = lnx  u dx
1
1 x
dx  
dx
x ln x
ln x
u'
  dx
u
 ln u  C
 ln ln x  C
Aim: Integrating Natural Log Function
u’ = 1/x
Divide N & D
by x
Substitute u
Log Rule
Back- substitute
Course: Calculus
Integrals of Trig Functions
Evaluate  tan x dx
sin x
tan x 
cos x
d
u'
cos x    sin x  dx

dx
u
sin x
 tan x dx   cos xdx
 sin x
 
dx
cos x
u'
   dx
u
 cos xdx  sin x  C
 sin xdx   cos x  C
 sec xdx  tan x  C
 sec x tan xdx  sec x  C
 csc xdx   cot x  C
2
2
u = cosx
u’ = -sinx
Substitute u
  ln u  C
Log Rule
  ln cos x  C
Back- substitute
Aim: Integrating Natural Log Function
Course: Calculus
Integrals of Trig Functions
Evaluate  sec x dx u 'dx  cos xdx  sin x  C
 u  sin xdx   cos x  C
sec x  tan x
Multiply by
sec x  tan x
2
sec
 xdx  tan x  C
 sec x tan xdx  sec x  C
 csc xdx   cot x  C
2
 sec x  tan x 
 sec x dx   sec x  sec x  tan x dx
 sec2 x  sec x tan x 
u'
 
dx   dx

u
 sec x  tan x 
u = sec x + tan x u’ = sec x tanx + sec2 x
 ln u  C
Log Rule
 ln sec x  tan x  C Back- substitute
Aim: Integrating Natural Log Function
Course: Calculus
Integrals for Basic Trig Functions
 sin u du   cos u  C
 cos u du  sin u  C
 tan u du   ln cos u  C
 cot u du  ln sin u  C
 sec u du  ln sec u  tan u  C
 csc u du   ln csc u  cot u  C
Aim: Integrating Natural Log Function
Course: Calculus
Model Problem
Evaluate

 4
1  tan 2 x dx
0
1 + tan2x = sec2x - Pythagorean Identity

 4
0
1  tan x dx =
2
=

 4
sec 2 x dx
0

 4
0
sec x dx
 sec u du  ln sec u  tan u  C
 4
 ln sec u  tan u  0
 ln


2  1  ln1
 0.8814
Aim: Integrating Natural Log Function
Course: Calculus
Model Problem
The electromotive force E of a particular
electrical circuit is given by E = 3sin2t,
where E is measured in volts and t is
measured in seconds. Find the average
value of E as t ranges from 0 to 0.5 second.
0.5
1
Average value =
3sin 2t dt

0.5  0 0
 6  sin 2t dt
0.5
0
u = 2t
du = 2dt
 1  0.5
 6     sin 2t  2  dt
 2 0
 3  cos 2t 0
0.5
Aim: Integrating Natural Log Function
 1.379 volts
Course: Calculus
Aim: How do we integrate the natural
logarithmic function?
Do Now:
Find the derivative
 4  x2 

y  ln 


x


Aim: Integrating Natural Log Function
Course: Calculus
Model Problem
Find the indefinite integral

 ln x 
x
2
dx
Aim: Integrating Natural Log Function
Course: Calculus