Download Ch 5: Integration Ch5.integration

137
§ 5.2 The Natural Log and Integration
Theorem 5.5: Log Rule for Integration
Let u be a differentiable function of x.
1
1.
 x dx  ln | x | c
2.
 u dx  ln | u | c
u'
Because the domain of ln does not include negative numbers we need to use
the absolute value in the antiderivative.
Example
3
i. Evaluate
 x dx
ii. Evaluate
 4 x  1 dx
1
138
iii. Evaluate using a change of variables.
2x
 ( x  1)2 dx 
iv. If a rational function has a numerator of degree greater than or equal to
that of the denominator, division may reveal a form to which we can apply the
Log Rule.
x2  x  1
 x2  1 dx 
§5.2: 9, 11, 18, 19
139
Guidelines for Integration
1. Memorize a basic list of the rules of integration. This includes the power
rule, the log rule and the trigonometric rules.
2. Find an integration formula that resembles all or part of the integrand.
Choose a satisfactory u by trial and error.
3. If you cannot find a satisfactory u BE CREATIVE! Try a trig identity,
multiplication and division of the same quantity etc.
4. Try using technology to help solve the antiderivative symbolically.
Example
v. Evaluate
vi.
1
 x ln x dx
 tan xdx
140
Integrals of the 6 Basic Trigonometric Functions
 sin udu   cos u  c
 cos udu  sin u  c
 tan udu   ln | cos u | c
 cot udu  ln | sin u | c
 sec udu  ln | sec u  tan u | c
 csc udu   ln | csc u  cot u | c
§5.2: 16, 22, 25, 27, 31, 34, 47
141
§ 5.4 Exponential Functions
3
2
1
-4
-2
2
4
6
-1
-2
-3
Consider the function f ( x)  ln x .
 Is increasing on its entire domain.
Thus
o f 1 ( x)  ln( f 1 ( x))  x If x is a real number.
But if x is a rational number,
f 1 ( x)  e x
Definition: The inverse of the natural log function f ( x)  ln x is called the
natural exponentiation function and is denoted by
f 1 ( x)  e x
That is
y  e x If and only if, x  ln y
142
Example
i. Solve 7  e
x 1
ln 2 x
 12
ii. Solve e
Theorem 5.10: Operations with Exponential Functions
Let a and b be real numbers.
e e e
a b
a b
ea
 e a b
b
e
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Theorem 5.11: Derivative of e
1.
d x
e   e x
dx
x
2.
d u
du
e   eu
dx
dx
Example
iii. Find
d
 e 2 x 1 
dx
iv. Find
d  3 x 
e

dx 
§5.4: 1, 4, 8, 9, 11, 14, 21-24, 35, 37, 40, 41, 43, 45, 57, 61, 63
144
x
Theorem 5.12: Integrating e
Let u be a differentiable function of x.
1.
 e dx  e
x
x
c
2.
Example
v. Evaluate
e
vi. Evaluate
 5xe
3 x 1
dx
 x2
dx
 e dx  e
u
u
c
145
vii. Evaluate
viii. Evaluate
1
e
0
x
dx
x
3
 e (4x )dx
4
146
e2 x
dx
ix. Evaluate 
1  e2 x
§5.4: 85, 87, 93, 95, 99
147
§ 5.5 Bases Other Than e and Applications
Definition of Exponential Function to Base a
If ‘a’ is a positive real number (a  1) and x is any real number, then the
exponential function to the base a is denoted a x and is defined by
a x  e(ln a ) x
If a  1 , then y  1x  1. (Constant Function)
Here are some familiar properties;
a0  1
a x a y  a x y
ax
 a x y
y
a
(a x ) y  a xy
Example
i. The half life of carbon-14 is about 5730 years. If one gram of carbon-14 is
present in a sample, how much will be in the sample in 10, 000 years?
148
Definition of Logarithmic function to Base a
‘a’ is a positive real number (a  1) and x is a positive real number, then the
logarithmic function to base a is denoted log a x and is defined by
log a x 
1
ln x
ln a
Here are some familiar properties;
log a 1  0
log a xy  log a x  log a y
log a
x
 log a x  log a y
y
log a x n  n log a x
y  a x if and only if x  log a y
a loga x  x, x  0
log a a x  x, for all x
Example
ii. Solve for x
5x 
1
625
§5.5: 5, 9, 12, 13, 17, 21
log3 x  4
149
Theorem 5.13: Derivatives for Bases Other Than e
Let ‘a’ be a positive real number (a  1) and let u be a differentiable function of
x
d
 a x   (ln a )a x
dx
d
du
 a x   (ln a )a u
dx
dx
d
1
log a x  
dx
(ln a ) x
d
1 du
log a u  
dx
(ln a )u dx
Example
iii. Find
dy
dx
f ( x)  4 x
§5.5: 38, 43, 46, 49, 53, 55
y  log 10 2 x
150
Occasionally, an integrand involves an exponential function to a base other
than e
1. Convert to base e
a x  e(ln a ) x
2.
 1 
 a dx   ln a  a
x
x
c
Example
iv. Evaluate
 6 dx
x
Theorem 5.14: Power Rule for Real Exponents
d n
 x   nx n 1
dx  
n= real number
d n
du
u   nu n 1
dx
dx
u= differentiable function of x
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Theorem 5.15: A Limit Involving e
 1
 x 1 
lim 1    lim 
 e
x 
 x  x  x 
x
x
Compound Interest Formulas
P= amount of deposit
t= number of years
A= balance after t years
r= annual interest rate (decimal form)
n= number of yearly compounding periods
Compounded n times per year
 r
A  P 1  
 n
nt
Compounded continuously
A  Pert
§5.5: 61, 63, 65, 69, 71, 79, 85-88
152
§ 5.7 Inverse Trig Functions and Differentiation
Consider one of the six trig functions:
sine, cosine, tangent, secant, cosecant, and cotangent
All are periodic. Thus none are one-to-one.
None of these six functions has an inverse!
However, on certain defined intervals, the six trig functions can be one-to-one.
On the interval   ,   the sine function is increasing and thus one-to-one.
 2 2
The inverse of the restricted sine function is
y  arcsin x if and only if sin y  x
Inverse Trig Functions
Function
y  arcsin x iff sin y  x
Domain
1  x  1
Range
  y  
2
2
y  arccos x iff cos y  x
1  x  1
0 y 
y  arctan x iff tan y  x
  x  

y  arccot x iff cot y  x
  x  
0 y 
y  arcsec x iff sec y  x
| x | 1
0 y  , y 
y  arc csc x iff csc y  x
| x | 1

2
2
 y 
2
2
 y  , y  0
2
iff= if and only if
** See the graphs of the six inverse trig functions on pg 371
153
Example
i. Solve for x
arctan(2 x  3)  
4
Theorem 5.17: Derivatives of Inverse Trig Functions
Let u be a differentiable function of x
d
u'
arcsin u  
dx
1 u2
d
u '
arccos u  
dx
1 u2
d
u'
arctan u   2
dx
1 u
d
u '
 arc cot u   2
dx
1 u
d
u'
 arc sec u  
dx
| u | u2 1
d
u '
 arc csc u  
dx
| u | u2 1
154
Example
ii. Find
iii. Find
d
arcsin 2 x  
dx
d
arctan 3x  
dx
155
iv. Find
d
arc sec e 2 x  
dx
**pg 376- Basic Differentiation Rules. This is a COMPLETE list of covered rules!
§ 5.7 1, 5, 7, 17, 19, 25-28, 43, 44, 48, 49, 53, 55