INSTRUCTORS Raffi Hovasapian John Zhu

Home » Mathematics » AP Calculus AB » Integration by Partial Fractions II

Raffi Hovasapian

Raffi Hovasapian

Integration by Partial Fractions II

Slide Duration:

Table of Contents

Section 1: Limits and Derivatives
Overview & Slopes of Curves

42m 8s

Intro
0:00
Overview & Slopes of Curves
0:21
Differential and Integral
0:22
Fundamental Theorem of Calculus
6:36
Differentiation or Taking the Derivative
14:24
What Does the Derivative Mean and How do We Find it?
15:18
Example: f'(x)
19:24
Example: f(x) = sin (x)
29:16
General Procedure for Finding the Derivative of f(x)
37:33
More on Slopes of Curves

50m 53s

Intro
0:00
Slope of the Secant Line along a Curve
0:12
Slope of the Tangent Line to f(x) at a Particlar Point
0:13
Slope of the Secant Line along a Curve
2:59
Instantaneous Slope
6:51
Instantaneous Slope
6:52
Example: Distance, Time, Velocity
13:32
Instantaneous Slope and Average Slope
25:42
Slope & Rate of Change
29:55
Slope & Rate of Change
29:56
Example: Slope = 2
33:16
Example: Slope = 4/3
34:32
Example: Slope = 4 (m/s)
39:12
Example: Density = Mass / Volume
40:33
Average Slope, Average Rate of Change, Instantaneous Slope, and Instantaneous Rate of Change
47:46
Example Problems for Slopes of Curves

59m 12s

Intro
0:00
Example I: Water Tank
0:13
Part A: Which is the Independent Variable and Which is the Dependent?
2:00
Part B: Average Slope
3:18
Part C: Express These Slopes as Rates-of-Change
9:28
Part D: Instantaneous Slope
14:54
Example II: y = √(x-3)
28:26
Part A: Calculate the Slope of the Secant Line
30:39
Part B: Instantaneous Slope
41:26
Part C: Equation for the Tangent Line
43:59
Example III: Object in the Air
49:37
Part A: Average Velocity
50:37
Part B: Instantaneous Velocity
55:30
Desmos Tutorial

18m 43s

Intro
0:00
Desmos Tutorial
1:42
Desmos Tutorial
1:43
Things You Must Learn To Do on Your Particular Calculator
2:39
Things You Must Learn To Do on Your Particular Calculator
2:40
Example I: y=sin x
4:54
Example II: y=x³ and y = d/(dx) (x³)
9:22
Example III: y = x² {-5 <= x <= 0} and y = cos x {0 < x < 6}
13:15
The Limit of a Function

51m 53s

Intro
0:00
The Limit of a Function
0:14
The Limit of a Function
0:15
Graph: Limit of a Function
12:24
Table of Values
16:02
lim x→a f(x) Does not Say What Happens When x = a
20:05
Example I: f(x) = x²
24:34
Example II: f(x) = 7
27:05
Example III: f(x) = 4.5
30:33
Example IV: f(x) = 1/x
34:03
Example V: f(x) = 1/x²
36:43
The Limit of a Function, Cont.
38:16
Infinity and Negative Infinity
38:17
Does Not Exist
42:45
Summary
46:48
Example Problems for the Limit of a Function

24m 43s

Intro
0:00
Example I: Explain in Words What the Following Symbols Mean
0:10
Example II: Find the Following Limit
5:21
Example III: Use the Graph to Find the Following Limits
7:35
Example IV: Use the Graph to Find the Following Limits
11:48
Example V: Sketch the Graph of a Function that Satisfies the Following Properties
15:25
Example VI: Find the Following Limit
18:44
Example VII: Find the Following Limit
20:06
Calculating Limits Mathematically

53m 48s

Intro
0:00
Plug-in Procedure
0:09
Plug-in Procedure
0:10
Limit Laws
9:14
Limit Law 1
10:05
Limit Law 2
10:54
Limit Law 3
11:28
Limit Law 4
11:54
Limit Law 5
12:24
Limit Law 6
13:14
Limit Law 7
14:38
Plug-in Procedure, Cont.
16:35
Plug-in Procedure, Cont.
16:36
Example I: Calculating Limits Mathematically
20:50
Example II: Calculating Limits Mathematically
27:37
Example III: Calculating Limits Mathematically
31:42
Example IV: Calculating Limits Mathematically
35:36
Example V: Calculating Limits Mathematically
40:58
Limits Theorem
44:45
Limits Theorem 1
44:46
Limits Theorem 2: Squeeze Theorem
46:34
Example VI: Calculating Limits Mathematically
49:26
Example Problems for Calculating Limits Mathematically

21m 22s

Intro
0:00
Example I: Evaluate the Following Limit by Showing Each Application of a Limit Law
0:16
Example II: Evaluate the Following Limit
1:51
Example III: Evaluate the Following Limit
3:36
Example IV: Evaluate the Following Limit
8:56
Example V: Evaluate the Following Limit
11:19
Example VI: Calculating Limits Mathematically
13:19
Example VII: Calculating Limits Mathematically
14:59
Calculating Limits as x Goes to Infinity

50m 1s

Intro
0:00
Limit as x Goes to Infinity
0:14
Limit as x Goes to Infinity
0:15
Let's Look at f(x) = 1 / (x-3)
1:04
Summary
9:34
Example I: Calculating Limits as x Goes to Infinity
12:16
Example II: Calculating Limits as x Goes to Infinity
21:22
Example III: Calculating Limits as x Goes to Infinity
24:10
Example IV: Calculating Limits as x Goes to Infinity
36:00
Example Problems for Limits at Infinity

36m 31s

Intro
0:00
Example I: Calculating Limits as x Goes to Infinity
0:14
Example II: Calculating Limits as x Goes to Infinity
3:27
Example III: Calculating Limits as x Goes to Infinity
8:11
Example IV: Calculating Limits as x Goes to Infinity
14:20
Example V: Calculating Limits as x Goes to Infinity
20:07
Example VI: Calculating Limits as x Goes to Infinity
23:36
Continuity

53m

Intro
0:00
Definition of Continuity
0:08
Definition of Continuity
0:09
Example: Not Continuous
3:52
Example: Continuous
4:58
Example: Not Continuous
5:52
Procedure for Finding Continuity
9:45
Law of Continuity
13:44
Law of Continuity
13:45
Example I: Determining Continuity on a Graph
15:55
Example II: Show Continuity & Determine the Interval Over Which the Function is Continuous
17:57
Example III: Is the Following Function Continuous at the Given Point?
22:42
Theorem for Composite Functions
25:28
Theorem for Composite Functions
25:29
Example IV: Is cos(x³ + ln x) Continuous at x=π/2?
27:00
Example V: What Value of A Will make the Following Function Continuous at Every Point of Its Domain?
34:04
Types of Discontinuity
39:18
Removable Discontinuity
39:33
Jump Discontinuity
40:06
Infinite Discontinuity
40:32
Intermediate Value Theorem
40:58
Intermediate Value Theorem: Hypothesis & Conclusion
40:59
Intermediate Value Theorem: Graphically
43:40
Example VI: Prove That the Following Function Has at Least One Real Root in the Interval [4,6]
47:46
Derivative I

40m 2s

Intro
0:00
Derivative
0:09
Derivative
0:10
Example I: Find the Derivative of f(x)=x³
2:20
Notations for the Derivative
7:32
Notations for the Derivative
7:33
Derivative & Rate of Change
11:14
Recall the Rate of Change
11:15
Instantaneous Rate of Change
17:04
Graphing f(x) and f'(x)
19:10
Example II: Find the Derivative of x⁴ - x²
24:00
Example III: Find the Derivative of f(x)=√x
30:51
Derivatives II

53m 45s

Intro
0:00
Example I: Find the Derivative of (2+x)/(3-x)
0:18
Derivatives II
9:02
f(x) is Differentiable if f'(x) Exists
9:03
Recall: For a Limit to Exist, Both Left Hand and Right Hand Limits Must Equal to Each Other
17:19
Geometrically: Differentiability Means the Graph is Smooth
18:44
Example II: Show Analytically that f(x) = |x| is Nor Differentiable at x=0
20:53
Example II: For x > 0
23:53
Example II: For x < 0
25:36
Example II: What is f(0) and What is the lim |x| as x→0?
30:46
Differentiability & Continuity
34:22
Differentiability & Continuity
34:23
How Can a Function Not be Differentiable at a Point?
39:38
How Can a Function Not be Differentiable at a Point?
39:39
Higher Derivatives
41:58
Higher Derivatives
41:59
Derivative Operator
45:12
Example III: Find (dy)/(dx) & (d²y)/(dx²) for y = x³
49:29
More Example Problems for The Derivative

31m 38s

Intro
0:00
Example I: Sketch f'(x)
0:10
Example II: Sketch f'(x)
2:14
Example III: Find the Derivative of the Following Function sing the Definition
3:49
Example IV: Determine f, f', and f'' on a Graph
12:43
Example V: Find an Equation for the Tangent Line to the Graph of the Following Function at the Given x-value
13:40
Example VI: Distance vs. Time
20:15
Example VII: Displacement, Velocity, and Acceleration
23:56
Example VIII: Graph the Displacement Function
28:20
Section 2: Differentiation
Differentiation of Polynomials & Exponential Functions

47m 35s

Intro
0:00
Differentiation of Polynomials & Exponential Functions
0:15
Derivative of a Function
0:16
Derivative of a Constant
2:35
Power Rule
3:08
If C is a Constant
4:19
Sum Rule
5:22
Exponential Functions
6:26
Example I: Differentiate
7:45
Example II: Differentiate
12:38
Example III: Differentiate
15:13
Example IV: Differentiate
16:20
Example V: Differentiate
19:19
Example VI: Find the Equation of the Tangent Line to a Function at a Given Point
12:18
Example VII: Find the First & Second Derivatives
25:59
Example VIII
27:47
Part A: Find the Velocity & Acceleration Functions as Functions of t
27:48
Part B: Find the Acceleration after 3 Seconds
30:12
Part C: Find the Acceleration when the Velocity is 0
30:53
Part D: Graph the Position, Velocity, & Acceleration Graphs
32:50
Example IX: Find a Cubic Function Whose Graph has Horizontal Tangents
34:53
Example X: Find a Point on a Graph
42:31
The Product, Power & Quotient Rules

47m 25s

Intro
0:00
The Product, Power and Quotient Rules
0:19
Differentiate Functions
0:20
Product Rule
5:30
Quotient Rule
9:15
Power Rule
10:00
Example I: Product Rule
13:48
Example II: Quotient Rule
16:13
Example III: Power Rule
18:28
Example IV: Find dy/dx
19:57
Example V: Find dy/dx
24:53
Example VI: Find dy/dx
28:38
Example VII: Find an Equation for the Tangent to the Curve
34:54
Example VIII: Find d²y/dx²
38:08
Derivatives of the Trigonometric Functions

41m 8s

Intro
0:00
Derivatives of the Trigonometric Functions
0:09
Let's Find the Derivative of f(x) = sin x
0:10
Important Limits to Know
4:59
d/dx (sin x)
6:06
d/dx (cos x)
6:38
d/dx (tan x)
6:50
d/dx (csc x)
7:02
d/dx (sec x)
7:15
d/dx (cot x)
7:27
Example I: Differentiate f(x) = x² - 4 cos x
7:56
Example II: Differentiate f(x) = x⁵ tan x
9:04
Example III: Differentiate f(x) = (cos x) / (3 + sin x)
10:56
Example IV: Differentiate f(x) = e^x / (tan x - sec x)
14:06
Example V: Differentiate f(x) = (csc x - 4) / (cot x)
15:37
Example VI: Find an Equation of the Tangent Line
21:48
Example VII: For What Values of x Does the Graph of the Function x + 3 cos x Have a Horizontal Tangent?
25:17
Example VIII: Ladder Problem
28:23
Example IX: Evaluate
33:22
Example X: Evaluate
36:38
The Chain Rule

24m 56s

Intro
0:00
The Chain Rule
0:13
Recall the Composite Functions
0:14
Derivatives of Composite Functions
1:34
Example I: Identify f(x) and g(x) and Differentiate
6:41
Example II: Identify f(x) and g(x) and Differentiate
9:47
Example III: Differentiate
11:03
Example IV: Differentiate f(x) = -5 / (x² + 3)³
12:15
Example V: Differentiate f(x) = cos(x² + c²)
14:35
Example VI: Differentiate f(x) = cos⁴x +c²
15:41
Example VII: Differentiate
17:03
Example VIII: Differentiate f(x) = sin(tan x²)
19:01
Example IX: Differentiate f(x) = sin(tan² x)
21:02
More Chain Rule Example Problems

25m 32s

Intro
0:00
Example I: Differentiate f(x) = sin(cos(tanx))
0:38
Example II: Find an Equation for the Line Tangent to the Given Curve at the Given Point
2:25
Example III: F(x) = f(g(x)), Find F' (6)
4:22
Example IV: Differentiate & Graph both the Function & the Derivative in the Same Window
5:35
Example V: Differentiate f(x) = ( (x-8)/(x+3) )⁴
10:18
Example VI: Differentiate f(x) = sec²(12x)
12:28
Example VII: Differentiate
14:41
Example VIII: Differentiate
19:25
Example IX: Find an Expression for the Rate of Change of the Volume of the Balloon with Respect to Time
21:13
Implicit Differentiation

52m 31s

Intro
0:00
Implicit Differentiation
0:09
Implicit Differentiation
0:10
Example I: Find (dy)/(dx) by both Implicit Differentiation and Solving Explicitly for y
12:15
Example II: Find (dy)/(dx) of x³ + x²y + 7y² = 14
19:18
Example III: Find (dy)/(dx) of x³y² + y³x² = 4x
21:43
Example IV: Find (dy)/(dx) of the Following Equation
24:13
Example V: Find (dy)/(dx) of 6sin x cos y = 1
29:00
Example VI: Find (dy)/(dx) of x² cos² y + y sin x = 2sin x cos y
31:02
Example VII: Find (dy)/(dx) of √(xy) = 7 + y²e^x
37:36
Example VIII: Find (dy)/(dx) of 4(x²+y²)² = 35(x²-y²)
41:03
Example IX: Find (d²y)/(dx²) of x² + y² = 25
44:05
Example X: Find (d²y)/(dx²) of sin x + cos y = sin(2x)
47:48
Section 3: Applications of the Derivative
Linear Approximations & Differentials

47m 34s

Intro
0:00
Linear Approximations & Differentials
0:09
Linear Approximations & Differentials
0:10
Example I: Linear Approximations & Differentials
11:27
Example II: Linear Approximations & Differentials
20:19
Differentials
30:32
Differentials
30:33
Example III: Linear Approximations & Differentials
34:09
Example IV: Linear Approximations & Differentials
35:57
Example V: Relative Error
38:46
Related Rates

45m 33s

Intro
0:00
Related Rates
0:08
Strategy for Solving Related Rates Problems #1
0:09
Strategy for Solving Related Rates Problems #2
1:46
Strategy for Solving Related Rates Problems #3
2:06
Strategy for Solving Related Rates Problems #4
2:50
Strategy for Solving Related Rates Problems #5
3:38
Example I: Radius of a Balloon
5:15
Example II: Ladder
12:52
Example III: Water Tank
19:08
Example IV: Distance between Two Cars
29:27
Example V: Line-of-Sight
36:20
More Related Rates Examples

37m 17s

Intro
0:00
Example I: Shadow
0:14
Example II: Particle
4:45
Example III: Water Level
10:28
Example IV: Clock
20:47
Example V: Distance between a House and a Plane
29:11
Maximum & Minimum Values of a Function

40m 44s

Intro
0:00
Maximum & Minimum Values of a Function, Part 1
0:23
Absolute Maximum
2:20
Absolute Minimum
2:52
Local Maximum
3:38
Local Minimum
4:26
Maximum & Minimum Values of a Function, Part 2
6:11
Function with Absolute Minimum but No Absolute Max, Local Max, and Local Min
7:18
Function with Local Max & Min but No Absolute Max & Min
8:48
Formal Definitions
10:43
Absolute Maximum
11:18
Absolute Minimum
12:57
Local Maximum
14:37
Local Minimum
16:25
Extreme Value Theorem
18:08
Theorem: f'(c) = 0
24:40
Critical Number (Critical Value)
26:14
Procedure for Finding the Critical Values of f(x)
28:32
Example I: Find the Critical Values of f(x) x + sinx
29:51
Example II: What are the Absolute Max & Absolute Minimum of f(x) = x + 4 sinx on [0,2π]
35:31
Example Problems for Max & Min

40m 44s

Intro
0:00
Example I: Identify Absolute and Local Max & Min on the Following Graph
0:11
Example II: Sketch the Graph of a Continuous Function
3:11
Example III: Sketch the Following Graphs
4:40
Example IV: Find the Critical Values of f (x) = 3x⁴ - 7x³ + 4x²
6:13
Example V: Find the Critical Values of f(x) = |2x - 5|
8:42
Example VI: Find the Critical Values
11:42
Example VII: Find the Critical Values f(x) = cos²(2x) on [0,2π]
16:57
Example VIII: Find the Absolute Max & Min f(x) = 2sinx + 2cos x on [0,(π/3)]
20:08
Example IX: Find the Absolute Max & Min f(x) = (ln(2x)) / x on [1,3]
24:39
The Mean Value Theorem

25m 54s

Intro
0:00
Rolle's Theorem
0:08
Rolle's Theorem: If & Then
0:09
Rolle's Theorem: Geometrically
2:06
There May Be More than 1 c Such That f'( c ) = 0
3:30
Example I: Rolle's Theorem
4:58
The Mean Value Theorem
9:12
The Mean Value Theorem: If & Then
9:13
The Mean Value Theorem: Geometrically
11:07
Example II: Mean Value Theorem
13:43
Example III: Mean Value Theorem
21:19
Using Derivatives to Graph Functions, Part I

25m 54s

Intro
0:00
Using Derivatives to Graph Functions, Part I
0:12
Increasing/ Decreasing Test
0:13
Example I: Find the Intervals Over Which the Function is Increasing & Decreasing
3:26
Example II: Find the Local Maxima & Minima of the Function
19:18
Example III: Find the Local Maxima & Minima of the Function
31:39
Using Derivatives to Graph Functions, Part II

44m 58s

Intro
0:00
Using Derivatives to Graph Functions, Part II
0:13
Concave Up & Concave Down
0:14
What Does This Mean in Terms of the Derivative?
6:14
Point of Inflection
8:52
Example I: Graph the Function
13:18
Example II: Function x⁴ - 5x²
19:03
Intervals of Increase & Decrease
19:04
Local Maxes and Mins
25:01
Intervals of Concavity & X-Values for the Points of Inflection
29:18
Intervals of Concavity & Y-Values for the Points of Inflection
34:18
Graphing the Function
40:52
Example Problems I

49m 19s

Intro
0:00
Example I: Intervals, Local Maxes & Mins
0:26
Example II: Intervals, Local Maxes & Mins
5:05
Example III: Intervals, Local Maxes & Mins, and Inflection Points
13:40
Example IV: Intervals, Local Maxes & Mins, Inflection Points, and Intervals of Concavity
23:02
Example V: Intervals, Local Maxes & Mins, Inflection Points, and Intervals of Concavity
34:36
Example Problems III

59m 1s

Intro
0:00
Example I: Intervals, Local Maxes & Mins, Inflection Points, Intervals of Concavity, and Asymptotes
0:11
Example II: Intervals, Local Maxes & Mins, Inflection Points, Intervals of Concavity, and Asymptotes
21:24
Example III: Cubic Equation f(x) = Ax³ + Bx² + Cx + D
37:56
Example IV: Intervals, Local Maxes & Mins, Inflection Points, Intervals of Concavity, and Asymptotes
46:19
L'Hospital's Rule

30m 9s

Intro
0:00
L'Hospital's Rule
0:19
Indeterminate Forms
0:20
L'Hospital's Rule
3:38
Example I: Evaluate the Following Limit Using L'Hospital's Rule
8:50
Example II: Evaluate the Following Limit Using L'Hospital's Rule
10:30
Indeterminate Products
11:54
Indeterminate Products
11:55
Example III: L'Hospital's Rule & Indeterminate Products
13:57
Indeterminate Differences
17:00
Indeterminate Differences
17:01
Example IV: L'Hospital's Rule & Indeterminate Differences
18:57
Indeterminate Powers
22:20
Indeterminate Powers
22:21
Example V: L'Hospital's Rule & Indeterminate Powers
25:13
Example Problems for L'Hospital's Rule

38m 14s

Intro
0:00
Example I: Evaluate the Following Limit
0:17
Example II: Evaluate the Following Limit
2:45
Example III: Evaluate the Following Limit
6:54
Example IV: Evaluate the Following Limit
8:43
Example V: Evaluate the Following Limit
11:01
Example VI: Evaluate the Following Limit
14:48
Example VII: Evaluate the Following Limit
17:49
Example VIII: Evaluate the Following Limit
20:37
Example IX: Evaluate the Following Limit
25:16
Example X: Evaluate the Following Limit
32:44
Optimization Problems I

49m 59s

Intro
0:00
Example I: Find the Dimensions of the Box that Gives the Greatest Volume
1:23
Fundamentals of Optimization Problems
18:08
Fundamental #1
18:33
Fundamental #2
19:09
Fundamental #3
19:19
Fundamental #4
20:59
Fundamental #5
21:55
Fundamental #6
23:44
Example II: Demonstrate that of All Rectangles with a Given Perimeter, the One with the Largest Area is a Square
24:36
Example III: Find the Points on the Ellipse 9x² + y² = 9 Farthest Away from the Point (1,0)
35:13
Example IV: Find the Dimensions of the Rectangle of Largest Area that can be Inscribed in a Circle of Given Radius R
43:10
Optimization Problems II

55m 10s

Intro
0:00
Example I: Optimization Problem
0:13
Example II: Optimization Problem
17:34
Example III: Optimization Problem
35:06
Example IV: Revenue, Cost, and Profit
43:22
Newton's Method

30m 22s

Intro
0:00
Newton's Method
0:45
Newton's Method
0:46
Example I: Find x2 and x3
13:18
Example II: Use Newton's Method to Approximate
15:48
Example III: Find the Root of the Following Equation to 6 Decimal Places
19:57
Example IV: Use Newton's Method to Find the Coordinates of the Inflection Point
23:11
Section 4: Integrals
Antiderivatives

55m 26s

Intro
0:00
Antiderivatives
0:23
Definition of an Antiderivative
0:24
Antiderivative Theorem
7:58
Function & Antiderivative
12:10
x^n
12:30
1/x
13:00
e^x
13:08
cos x
13:18
sin x
14:01
sec² x
14:11
secxtanx
14:18
1/√(1-x²)
14:26
1/(1+x²)
14:36
-1/√(1-x²)
14:45
Example I: Find the Most General Antiderivative for the Following Functions
15:07
Function 1: f(x) = x³ -6x² + 11x - 9
15:42
Function 2: f(x) = 14√(x) - 27 4√x
19:12
Function 3: (fx) = cos x - 14 sinx
20:53
Function 4: f(x) = (x⁵+2√x )/( x^(4/3) )
22:10
Function 5: f(x) = (3e^x) - 2/(1+x²)
25:42
Example II: Given the Following, Find the Original Function f(x)
26:37
Function 1: f'(x) = 5x³ - 14x + 24, f(2) = 40
27:55
Function 2: f'(x) 3 sinx + sec²x, f(π/6) = 5
30:34
Function 3: f''(x) = 8x - cos x, f(1.5) = 12.7, f'(1.5) = 4.2
32:54
Function 4: f''(x) = 5/(√x), f(2) 15, f'(2) = 7
37:54
Example III: Falling Object
41:58
Problem 1: Find an Equation for the Height of the Ball after t Seconds
42:48
Problem 2: How Long Will It Take for the Ball to Strike the Ground?
48:30
Problem 3: What is the Velocity of the Ball as it Hits the Ground?
49:52
Problem 4: Initial Velocity of 6 m/s, How Long Does It Take to Reach the Ground?
50:46
The Area Under a Curve

51m 3s

Intro
0:00
The Area Under a Curve
0:13
Approximate Using Rectangles
0:14
Let's Do This Again, Using 4 Different Rectangles
9:40
Approximate with Rectangles
16:10
Left Endpoint
18:08
Right Endpoint
25:34
Left Endpoint vs. Right Endpoint
30:58
Number of Rectangles
34:08
True Area
37:36
True Area
37:37
Sigma Notation & Limits
43:32
When You Have to Explicitly Solve Something
47:56
Example Problems for Area Under a Curve

33m 7s

Intro
0:00
Example I: Using Left Endpoint & Right Endpoint to Approximate Area Under a Curve
0:10
Example II: Using 5 Rectangles, Approximate the Area Under the Curve
11:32
Example III: Find the True Area by Evaluating the Limit Expression
16:07
Example IV: Find the True Area by Evaluating the Limit Expression
24:52
The Definite Integral

43m 19s

Intro
0:00
The Definite Integral
0:08
Definition to Find the Area of a Curve
0:09
Definition of the Definite Integral
4:08
Symbol for Definite Integral
8:45
Regions Below the x-axis
15:18
Associating Definite Integral to a Function
19:38
Integrable Function
27:20
Evaluating the Definite Integral
29:26
Evaluating the Definite Integral
29:27
Properties of the Definite Integral
35:24
Properties of the Definite Integral
35:25
Example Problems for The Definite Integral

32m 14s

Intro
0:00
Example I: Approximate the Following Definite Integral Using Midpoints & Sub-intervals
0:11
Example II: Express the Following Limit as a Definite Integral
5:28
Example III: Evaluate the Following Definite Integral Using the Definition
6:28
Example IV: Evaluate the Following Integral Using the Definition
17:06
Example V: Evaluate the Following Definite Integral by Using Areas
25:41
Example VI: Definite Integral
30:36
The Fundamental Theorem of Calculus

24m 17s

Intro
0:00
The Fundamental Theorem of Calculus
0:17
Evaluating an Integral
0:18
Lim as x → ∞
12:19
Taking the Derivative
14:06
Differentiation & Integration are Inverse Processes
15:04
1st Fundamental Theorem of Calculus
20:08
1st Fundamental Theorem of Calculus
20:09
2nd Fundamental Theorem of Calculus
22:30
2nd Fundamental Theorem of Calculus
22:31
Example Problems for the Fundamental Theorem

25m 21s

Intro
0:00
Example I: Find the Derivative of the Following Function
0:17
Example II: Find the Derivative of the Following Function
1:40
Example III: Find the Derivative of the Following Function
2:32
Example IV: Find the Derivative of the Following Function
5:55
Example V: Evaluate the Following Integral
7:13
Example VI: Evaluate the Following Integral
9:46
Example VII: Evaluate the Following Integral
12:49
Example VIII: Evaluate the Following Integral
13:53
Example IX: Evaluate the Following Graph
15:24
Local Maxs and Mins for g(x)
15:25
Where Does g(x) Achieve Its Absolute Max on [0,8]
20:54
On What Intervals is g(x) Concave Up/Down?
22:20
Sketch a Graph of g(x)
24:34
More Example Problems, Including Net Change Applications

34m 22s

Intro
0:00
Example I: Evaluate the Following Indefinite Integral
0:10
Example II: Evaluate the Following Definite Integral
0:59
Example III: Evaluate the Following Integral
2:59
Example IV: Velocity Function
7:46
Part A: Net Displacement
7:47
Part B: Total Distance Travelled
13:15
Example V: Linear Density Function
20:56
Example VI: Acceleration Function
25:10
Part A: Velocity Function at Time t
25:11
Part B: Total Distance Travelled During the Time Interval
28:38
Solving Integrals by Substitution

27m 20s

Intro
0:00
Table of Integrals
0:35
Example I: Evaluate the Following Indefinite Integral
2:02
Example II: Evaluate the Following Indefinite Integral
7:27
Example IIII: Evaluate the Following Indefinite Integral
10:57
Example IV: Evaluate the Following Indefinite Integral
12:33
Example V: Evaluate the Following
14:28
Example VI: Evaluate the Following
16:00
Example VII: Evaluate the Following
19:01
Example VIII: Evaluate the Following
21:49
Example IX: Evaluate the Following
24:34
Section 5: Applications of Integration
Areas Between Curves

34m 56s

Intro
0:00
Areas Between Two Curves: Function of x
0:08
Graph 1: Area Between f(x) & g(x)
0:09
Graph 2: Area Between f(x) & g(x)
4:07
Is It Possible to Write as a Single Integral?
8:20
Area Between the Curves on [a,b]
9:24
Absolute Value
10:32
Formula for Areas Between Two Curves: Top Function - Bottom Function
17:03
Areas Between Curves: Function of y
17:49
What if We are Given Functions of y?
17:50
Formula for Areas Between Two Curves: Right Function - Left Function
21:48
Finding a & b
22:32
Example Problems for Areas Between Curves

42m 55s

Intro
0:00
Instructions for the Example Problems
0:10
Example I: y = 7x - x² and y=x
0:37
Example II: x=y²-3, x=e^((1/2)y), y=-1, and y=2
6:25
Example III: y=(1/x), y=(1/x³), and x=4
12:25
Example IV: 15-2x² and y=x²-5
15:52
Example V: x=(1/8)y³ and x=6-y²
20:20
Example VI: y=cos x, y=sin(2x), [0,π/2]
24:34
Example VII: y=2x², y=10x², 7x+2y=10
29:51
Example VIII: Velocity vs. Time
33:23
Part A: At 2.187 Minutes, Which care is Further Ahead?
33:24
Part B: If We Shaded the Region between the Graphs from t=0 to t=2.187, What Would This Shaded Area Represent?
36:32
Part C: At 4 Minutes Which Car is Ahead?
37:11
Part D: At What Time Will the Cars be Side by Side?
37:50
Volumes I: Slices

34m 15s

Intro
0:00
Volumes I: Slices
0:18
Rotate the Graph of y=√x about the x-axis
0:19
How can I use Integration to Find the Volume?
3:16
Slice the Solid Like a Loaf of Bread
5:06
Volumes Definition
8:56
Example I: Find the Volume of the Solid Obtained by Rotating the Region Bounded by the Given Functions about the Given Line of Rotation
12:18
Example II: Find the Volume of the Solid Obtained by Rotating the Region Bounded by the Given Functions about the Given Line of Rotation
19:05
Example III: Find the Volume of the Solid Obtained by Rotating the Region Bounded by the Given Functions about the Given Line of Rotation
25:28
Volumes II: Volumes by Washers

51m 43s

Intro
0:00
Volumes II: Volumes by Washers
0:11
Rotating Region Bounded by y=x³ & y=x around the x-axis
0:12
Equation for Volumes by Washer
11:14
Process for Solving Volumes by Washer
13:40
Example I: Find the Volume of the Solid Obtained by Rotating the Region Bounded by the Following Functions around the Given Axis
15:58
Example II: Find the Volume of the Solid Obtained by Rotating the Region Bounded by the Following Functions around the Given Axis
25:07
Example III: Find the Volume of the Solid Obtained by Rotating the Region Bounded by the Following Functions around the Given Axis
34:20
Example IV: Find the Volume of the Solid Obtained by Rotating the Region Bounded by the Following Functions around the Given Axis
44:05
Volumes III: Solids That Are Not Solids-of-Revolution

49m 36s

Intro
0:00
Solids That Are Not Solids-of-Revolution
0:11
Cross-Section Area Review
0:12
Cross-Sections That Are Not Solids-of-Revolution
7:36
Example I: Find the Volume of a Pyramid Whose Base is a Square of Side-length S, and Whose Height is H
10:54
Example II: Find the Volume of a Solid Whose Cross-sectional Areas Perpendicular to the Base are Equilateral Triangles
20:39
Example III: Find the Volume of a Pyramid Whose Base is an Equilateral Triangle of Side-Length A, and Whose Height is H
29:27
Example IV: Find the Volume of a Solid Whose Base is Given by the Equation 16x² + 4y² = 64
36:47
Example V: Find the Volume of a Solid Whose Base is the Region Bounded by the Functions y=3-x² and the x-axis
46:13
Volumes IV: Volumes By Cylindrical Shells

50m 2s

Intro
0:00
Volumes by Cylindrical Shells
0:11
Find the Volume of the Following Region
0:12
Volumes by Cylindrical Shells: Integrating Along x
14:12
Volumes by Cylindrical Shells: Integrating Along y
14:40
Volumes by Cylindrical Shells Formulas
16:22
Example I: Using the Method of Cylindrical Shells, Find the Volume of the Solid
18:33
Example II: Using the Method of Cylindrical Shells, Find the Volume of the Solid
25:57
Example III: Using the Method of Cylindrical Shells, Find the Volume of the Solid
31:38
Example IV: Using the Method of Cylindrical Shells, Find the Volume of the Solid
38:44
Example V: Using the Method of Cylindrical Shells, Find the Volume of the Solid
44:03
The Average Value of a Function

32m 13s

Intro
0:00
The Average Value of a Function
0:07
Average Value of f(x)
0:08
What if The Domain of f(x) is Not Finite?
2:23
Let's Calculate Average Value for f(x) = x² [2,5]
4:46
Mean Value Theorem for Integrate
9:25
Example I: Find the Average Value of the Given Function Over the Given Interval
14:06
Example II: Find the Average Value of the Given Function Over the Given Interval
18:25
Example III: Find the Number A Such that the Average Value of the Function f(x) = -4x² + 8x + 4 Equals 2 Over the Interval [-1,A]
24:04
Example IV: Find the Average Density of a Rod
27:47
Section 6: Techniques of Integration
Integration by Parts

50m 32s

Intro
0:00
Integration by Parts
0:08
The Product Rule for Differentiation
0:09
Integrating Both Sides Retains the Equality
0:52
Differential Notation
2:24
Example I: ∫ x cos x dx
5:41
Example II: ∫ x² sin(2x)dx
12:01
Example III: ∫ (e^x) cos x dx
18:19
Example IV: ∫ (sin^-1) (x) dx
23:42
Example V: ∫₁⁵ (lnx)² dx
28:25
Summary
32:31
Tabular Integration
35:08
Case 1
35:52
Example: ∫x³sinx dx
36:39
Case 2
40:28
Example: ∫e^(2x) sin 3x
41:14
Trigonometric Integrals I

24m 50s

Intro
0:00
Example I: ∫ sin³ (x) dx
1:36
Example II: ∫ cos⁵(x)sin²(x)dx
4:36
Example III: ∫ sin⁴(x)dx
9:23
Summary for Evaluating Trigonometric Integrals of the Following Type: ∫ (sin^m) (x) (cos^p) (x) dx
15:59
#1: Power of sin is Odd
16:00
#2: Power of cos is Odd
16:41
#3: Powers of Both sin and cos are Odd
16:55
#4: Powers of Both sin and cos are Even
17:10
Example IV: ∫ tan⁴ (x) sec⁴ (x) dx
17:34
Example V: ∫ sec⁹(x) tan³(x) dx
20:55
Summary for Evaluating Trigonometric Integrals of the Following Type: ∫ (sec^m) (x) (tan^p) (x) dx
23:31
#1: Power of sec is Odd
23:32
#2: Power of tan is Odd
24:04
#3: Powers of sec is Odd and/or Power of tan is Even
24:18
Trigonometric Integrals II

22m 12s

Intro
0:00
Trigonometric Integrals II
0:09
Recall: ∫tanx dx
0:10
Let's Find ∫secx dx
3:23
Example I: ∫ tan⁵ (x) dx
6:23
Example II: ∫ sec⁵ (x) dx
11:41
Summary: How to Deal with Integrals of Different Types
19:04
Identities to Deal with Integrals of Different Types
19:05
Example III: ∫cos(5x)sin(9x)dx
19:57
More Example Problems for Trigonometric Integrals

17m 22s

Intro
0:00
Example I: ∫sin²(x)cos⁷(x)dx
0:14
Example II: ∫x sin²(x) dx
3:56
Example III: ∫csc⁴ (x/5)dx
8:39
Example IV: ∫( (1-tan²x)/(sec²x) ) dx
11:17
Example V: ∫ 1 / (sinx-1) dx
13:19
Integration by Partial Fractions I

55m 12s

Intro
0:00
Integration by Partial Fractions I
0:11
Recall the Idea of Finding a Common Denominator
0:12
Decomposing a Rational Function to Its Partial Fractions
4:10
2 Types of Rational Function: Improper & Proper
5:16
Improper Rational Function
7:26
Improper Rational Function
7:27
Proper Rational Function
11:16
Proper Rational Function & Partial Fractions
11:17
Linear Factors
14:04
Irreducible Quadratic Factors
15:02
Case 1: G(x) is a Product of Distinct Linear Factors
17:10
Example I: Integration by Partial Fractions
20:33
Case 2: D(x) is a Product of Linear Factors
40:58
Example II: Integration by Partial Fractions
44:41
Integration by Partial Fractions II

42m 57s

Intro
0:00
Case 3: D(x) Contains Irreducible Factors
0:09
Example I: Integration by Partial Fractions
5:19
Example II: Integration by Partial Fractions
16:22
Case 4: D(x) has Repeated Irreducible Quadratic Factors
27:30
Example III: Integration by Partial Fractions
30:19
Section 7: Differential Equations
Introduction to Differential Equations

46m 37s

Intro
0:00
Introduction to Differential Equations
0:09
Overview
0:10
Differential Equations Involving Derivatives of y(x)
2:08
Differential Equations Involving Derivatives of y(x) and Function of y(x)
3:23
Equations for an Unknown Number
6:28
What are These Differential Equations Saying?
10:30
Verifying that a Function is a Solution of the Differential Equation
13:00
Verifying that a Function is a Solution of the Differential Equation
13:01
Verify that y(x) = 4e^x + 3x² + 6x + e^π is a Solution of this Differential Equation
17:20
General Solution
22:00
Particular Solution
24:36
Initial Value Problem
27:42
Example I: Verify that a Family of Functions is a Solution of the Differential Equation
32:24
Example II: For What Values of K Does the Function Satisfy the Differential Equation
36:07
Example III: Verify the Solution and Solve the Initial Value Problem
39:47
Separation of Variables

28m 8s

Intro
0:00
Separation of Variables
0:28
Separation of Variables
0:29
Example I: Solve the Following g Initial Value Problem
8:29
Example II: Solve the Following g Initial Value Problem
13:46
Example III: Find an Equation of the Curve
18:48
Population Growth: The Standard & Logistic Equations

51m 7s

Intro
0:00
Standard Growth Model
0:30
Definition of the Standard/Natural Growth Model
0:31
Initial Conditions
8:00
The General Solution
9:16
Example I: Standard Growth Model
10:45
Logistic Growth Model
18:33
Logistic Growth Model
18:34
Solving the Initial Value Problem
25:21
What Happens When t → ∞
36:42
Example II: Solve the Following g Initial Value Problem
41:50
Relative Growth Rate
46:56
Relative Growth Rate
46:57
Relative Growth Rate Version for the Standard model
49:04
Slope Fields

24m 37s

Intro
0:00
Slope Fields
0:35
Slope Fields
0:36
Graphing the Slope Fields, Part 1
11:12
Graphing the Slope Fields, Part 2
15:37
Graphing the Slope Fields, Part 3
17:25
Steps to Solving Slope Field Problems
20:24
Example I: Draw or Generate the Slope Field of the Differential Equation y'=x cos y
22:38
Section 8: AP Practic Exam
AP Practice Exam: Section 1, Part A No Calculator

45m 29s

Intro
0:00
Exam Link
0:10
Problem #1
1:26
Problem #2
2:52
Problem #3
4:42
Problem #4
7:03
Problem #5
10:01
Problem #6
13:49
Problem #7
15:16
Problem #8
19:06
Problem #9
23:10
Problem #10
28:10
Problem #11
31:30
Problem #12
33:53
Problem #13
37:45
Problem #14
41:17
AP Practice Exam: Section 1, Part A No Calculator, cont.

41m 55s

Intro
0:00
Problem #15
0:22
Problem #16
3:10
Problem #17
5:30
Problem #18
8:03
Problem #19
9:53
Problem #20
14:51
Problem #21
17:30
Problem #22
22:12
Problem #23
25:48
Problem #24
29:57
Problem #25
33:35
Problem #26
35:57
Problem #27
37:57
Problem #28
40:04
AP Practice Exam: Section I, Part B Calculator Allowed

58m 47s

Intro
0:00
Problem #1
1:22
Problem #2
4:55
Problem #3
10:49
Problem #4
13:05
Problem #5
14:54
Problem #6
17:25
Problem #7
18:39
Problem #8
20:27
Problem #9
26:48
Problem #10
28:23
Problem #11
34:03
Problem #12
36:25
Problem #13
39:52
Problem #14
43:12
Problem #15
47:18
Problem #16
50:41
Problem #17
56:38
AP Practice Exam: Section II, Part A Calculator Allowed

25m 40s

Intro
0:00
Problem #1: Part A
1:14
Problem #1: Part B
4:46
Problem #1: Part C
8:00
Problem #2: Part A
12:24
Problem #2: Part B
16:51
Problem #2: Part C
17:17
Problem #3: Part A
18:16
Problem #3: Part B
19:54
Problem #3: Part C
21:44
Problem #3: Part D
22:57
AP Practice Exam: Section II, Part B No Calculator

31m 20s

Intro
0:00
Problem #4: Part A
1:35
Problem #4: Part B
5:54
Problem #4: Part C
8:50
Problem #4: Part D
9:40
Problem #5: Part A
11:26
Problem #5: Part B
13:11
Problem #5: Part C
15:07
Problem #5: Part D
19:57
Problem #6: Part A
22:01
Problem #6: Part B
25:34
Problem #6: Part C
28:54
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Integration by Partial Fractions II

Lecture Slides are screen-captured images of important points in the lecture. Students can download and print out these lecture slide images to do practice problems as well as take notes while watching the lecture.

  • Intro 0:00
  • Case 3: D(x) Contains Irreducible Factors 0:09
  • Example I: Integration by Partial Fractions 5:19
  • Example II: Integration by Partial Fractions 16:22
  • Case 4: D(x) has Repeated Irreducible Quadratic Factors 27:30
  • Example III: Integration by Partial Fractions 30:19

Transcription: Integration by Partial Fractions II

Hello, welcome back to www.educator.com, and welcome back to AP Calculus.0000

Today, we are going to continue our discussion of integration by partial fractions.0004

Let us jump right on in.0008

If you remember in the last lesson, we did the first two of the four cases,0011

where once we factored the denominator of the rational expression as much as we can,0016

if we have all linear factors or if some of those linear factors are repeated.0023

Those are the first two cases that we dealt with last time.0028

Now in this lesson, we are going to talk about quadratic factors and repeated quadratic factors, the last two cases.0031

Let us start with case 3, I think I’m going to go ahead and work in blue.0039

Case 3 is when our rational function, some m(x)/d(x),0044

Again, this is just numerator and denominator as functions of x.0051

That is when the d(x) contains irreducible quadratic factors.0055

After you reduce, after you factored it as much as you can, one of the factors is quadratic.0064

In other words, the highest degree on the x, the highest exponent is a 2,0068

contains irreducible factors none of which is repeated.0074

The first one is going to be none repeating irreducible quadratic factors is repeated.0090

Let us recall irreducible means we cannot factor it any further.0103

We cannot factor the quadratic any further into linear factors.0115

That is our hope, we want to be able to factor something, any polynomial into, as many linear factors as possible.0134

As it turns out, ultimately, you can only go to linear and quadratic.0141

You can always you do that, but sometimes you cannot always factor the quadratic.0145

That is the irreducible.0149

Here is an example of irreducible is x² + 4.0151

This one, you cannot reduce any further.0161

Another example might be something like x² + 3x + 1.0164

You cannot factor it any further.0172

We said in the previous lesson that each linear factor ax + b gives a partial fraction of a/ ax + b.0177

If one of the factors is linear on top, the variable, the thing that we are looking for is just a constant, it is just a.0212

Now for quadratic factors, each factor of the form ax² + bx + c, because the quadratic factor is going to be some variation of this.0219

Sometimes the bx term would not be there.0248

Sometimes the c would not be there.0250

It is going to look something like that.0255

It gives ax + b/ ax² + bx + c.0257

On the top, our unknown, where it is actually going to be a full linear factor.0267

The thing to notice here is the denominators of degree 1, the numerator is 1° less.0273

That is why it is just a, it is x⁰.0278

Here the denominator, the highest degree is 2.0281

Therefore, on the numerator, it is going to be 1° less which means this type of function ax + b, a linear factor.0285

For linear factors, we put a on top, a constant.0295

For quadratic factors, we put ax + b.0298

Sorry, we use capitals, ax + b.0301

Where a, b, and c, and d, it could be, if you have another quadratic factor, it would be cx + d, ex + f, and so on.0304

Let us go ahead and do an example, I think it will make sense.0315

We want to evaluate the integral 12/ x - 2 × x² + 9.0321

The denominator here, again, the first thing we do is factor the denominator as much as we can.0326

Here the denominators are already factored.0331

x - 2 is our linear factor, x² + 9 is our irreducible quadratic factor.0333

This does not factor anymore.0338

If it were x² – 9, that is fine, we can do x + 3x - 3 but were stuck like this.0339

The denominators are already factored.0346

Now 12, this 12/ x - 2 × x² + 9, it is going to equal, we have a linear factor x – 2.0358

The partial fraction decomposition is going to be a/ x - 2 + this is our quadratic factor.0373

This is going to be x² + 9 and we put bx + c.0381

Our task is to find the a, find the b, and find the c, so that we have a partial fraction decomposition of our original rational function.0386

Once we separate that, we are going to integrate each one separately.0395

Let us go ahead and do that.0398

This is going to be, I actually solve for the least common denominator here.0407

This is going to be a × x² + 9 + bx + c × x - 2/ the least common denominator which is x - 2 × x² + 9.0411

We are going to concern ourselves only with the numerator.0433

Because now we have this equal to this, the denominators, this and this are the same,0435

that means that the numerator and the numerator are the same.0447

I'm just going to work with the numerators.0451

Once you actually do that on the right, once you find a common denominator, put it under a common denominator.0454

The denominators go away.0460

They are equal, therefore, the numerators are equal.0461

We concern ourselves only with the numerators.0464

Therefore, we have 12 is equal to, I can multiply this all this out here.0476

We have ax² + 9a + bx² + cx - 2bx – 2c.0482

12 =, I’m going to take care of the ax² bx².0506

It is going to be x² × a + b, that takes care of the ax² + bx².0510

We will do the x terms.0518

The x terms, I have c - 2b that takes care of the x terms.0520

I have + 9a - 2c, 9a - 2c that takes care of the number terms.0528

Now I equate coefficients.0540

Over on the left, there is no x² term.0542

Therefore, a + b is equal to 0, 0x².0545

Therefore, I have the equation, a + b is equal to 0.0549

On the left, there is no x term, therefore, it is 0x.0553

Therefore, c - 2b is equal to 0.0557

The number 12 is that one.0561

Therefore, I have 9a - 2c is equal to 12.0565

These three, these three equations and three unknowns is what we are going to solve for a, b, and c.0571

Let us go ahead and do that next.0576

I have got a + b = 0, I’m going to write them this way.0582

C - 2b = 0 and 9a - 2c = 12.0586

I presume that most of you are comfortable with solving two and three, sometimes four equations, and that many unknowns.0596

But I will go through the process anyway, it is not a problem.0603

It only takes a couple of minutes.0605

Here a = -b, here c = 2b.0607

I’m going to put this a and this c into here and solve for b, and then put the b’s back and find a and c.0614

I have 9 × a which is - b - 2 × 2b which is c equal to 12.0621

I have -9b - 4b = 12.0633

Sorry, this looks like a 13, this is a b.0641

I have got -13b is equal to 12.0644

Therefore, I find that b is equal to -12/13.0648

That takes care of b.0654

Now I go ahead and put that over here.0655

I find that a = - a -12/13 which means that a is equal to 12/13.0657

Of course, c is equal of 2 × b which is -12/13.0669

Therefore, c is equal to -24/13.0675

Now I found a, b, and c, I put them back into my original decomposition.0680

Remember, we had that our original 12/ x - 2 × x² + 9 is equal to, our decomposition was a/ x - 2 + bx + c/ x² + 9.0686

Therefore, we just put it in.0706

I have a is 12/13, this is going to be 12/13 / x - 2 + b which is -12/13 x + c which is -24/13 / x² + 9.0707

There you go, this is our partial fraction decomposition.0735

This is the first part, we did our partial fraction decomposition.0742

Now we actually want to integrate this.0745

The integral of this is going to be the integral of this.0747

The integral of this is going to equal the integral of this + the integral of that.0751

That is it, just work your way through.0755

Let us see what we have got here.0759

Our integral, which I will just call int, is equal to the integral of 12/13 / x - 2 dx0766

+ the integral of -12/13 x - 24/13 / x² + 9 dx, which is going to end up equaling 12/130779

× the integral of dx/ x - 2 - 12/13 × the integral of x/ x² + 9.0809

Just separating this thing out -24/13 the integral of, this is x dx, sorry about that.0824

I always forget the dx, for all these years, I still forget it.0836

x² + 9.0841

We end up getting the following.0846

We end up getting 12/13 × the natlog of the absolute value of x - 2 - 12/13 × ½ the natlog of x² + 9.0847

This ½ factor came from the fact that this is a u substitution.0866

I let u equal x², therefore, du = 2x dx x dx.0870

Bring the 2, u substitution, I will let you work that out.0876

This one is going to be -24/13 √9.0882

I will tell you where this came from in just a minute.0890

1x/ √9 + c, there we go.0894

The last integral, this one right here, where did I get that?0907

Here is where I got that.0914

For the last integral, we use the following formula.0916

The integral of 1/ x² + a² dx is equal to 1/a × tan⁻¹(x)/ a.0927

That is the form that we use for this because often, when we do partial fraction decompositions,0943

especially when we have quadratic factors in the denominator, we often end up with integrals that look like that.0947

Some dx which is just the 1 dx over here/ something x² + something else x² + something else².0953

It is the general integral that keep showing up.0966

We went ahead and we are just going to use the formula for it.0970

1/a × tan⁻¹(x)/ a.0972

Let us do another example here, this time we want to evaluate the integral of x – 4/ 4x² + 4x +5.0981

We take a look at this and we realize that the denominator cannot be factored any further.0990

This is definitely an irreducible quadratic factor.1007

4x² - 4x + 5 is irreducible and it is also the only factor.1014

Because it is the only factor, there is no partial fraction decomposition.1039

The rational function itself, it is the partial fraction decomposition.1043

It is a partial fraction decomposition that is composed of just one term, x - 4/ 4x² – 4x + 5.1048

There are no other factors in the denominator, for me to actually expand and do what I have done in the previous problems.1055

Here we are going to show you a general procedure for how to handle a quadratic in the denominator that is the only term.1061

Here is how we do it.1070

We actually are going to complete the square in the denominator.1074

We handle the situation, again, this is a general procedure,1079

anytime you have a quadratic in the denominator that it is the only factor.1082

It is a single only factor.1086

We handle this situation by completing the square.1089

Something that you have done thousands of times in algebra, by completing the square in the denominator.1096

I got to tell you the technique of completing the square is something that comes in handy so often,1107

and so many other branches of mathematics.1114

We are working just with the denominator.1120

The denominator, we got 4x² - 4x + 5.1125

I’m going to go ahead and factor out the 4.1134

This is going to be 4 × x² – x.1136

I will leave a little space for something that I add, + 5.1142

I take half of the second term which is -1/2 and I square it.1146

This is going to be + ¼.1149

Since I added 4 × 1/4, I added 1, I'm going to subtract 1 from this expression to retain the equality.1156

I'm going to write this as 4 × x – ½² + 4.1164

This is just 2² × x – ½² + 4.1174

Again, this is just mathematical manipulation, nothing strange happening here.1181

This is 2² × something², I’m going to put them together and take the squared out.1185

This is going to be 2 × x - ½² + 4.1191

I'm going to multiply, I’m going to distribute the 2 in there.1202

This is going to end up being 2x - 1² + 4.1204

Now I have that, this is my denominator.1217

I just changed the way it looks.1226

We have the integral of x - 4/ 2x - 1² + 4 dx.1235

Now I’m going to subject this to a u substitution.1247

Let us do this in red.1254

I’m going to let u equal 2x – 1.1255

I’m going to let du = 2 dx, that means dx is equal to du/2.1263

I have taken care of that.1274

Over here I’m going to actually solve for x.1278

It is going to be x is going to equal, because I want to also deal with this x on top, in terms of u.1280

x, when I solve this equation, we said u = 2x - 1 so x = u + 1/ 2.1291

When I put all of these in here, we have the integral of u + 1.1299

I’m going to write it as ½ of u + 1, that is my x, - 4/ u² + 4 dx is du/2 which = ½.1316

I’m going to pull this ½ out.1338

The integral of ½ u + ½ - 4 which I’m going to write as 8/2 / u² + 4 du1339

= ½ of the integral of ½ u - 7/2 / u² + 4 du.1363

I’m going to pull the ½ here, put it here, it equals ½ × ½ the integral of u - 7/ u² + 4.1374

So far so good, that = ¼ × the integral of u/ u² + 4 du - 7/4 × the integral of 1/ u² + 4 du.1389

Now these integrals, I can handle.1413

Remember, again, u = 2x – 1.1416

Let us actually write this again.1425

We said that it equal ¼ × the integral of u/ u² + 4 du – 7/4 × the integral of 1/ u² + 4 du.1437

That gives us ¼ × ½ × natlog of u² + u – 7/4 × ½ tan⁻¹ of u/ 2 + c.1455

This ½ term, that one comes from the fact that I do a second u substitution on this.1485

If I call that one v, v = u².1494

This ½ comes from the fact that this is that formula, it is 1/a u² + 2².1502

Remember that formula that we just did.1511

We said that the integral of 1/ u² + a² = 1/a × tan⁻¹ of 1/a.1515

Here this is a² which means that a is 2.1525

That is where this one actually comes from.1530

Sorry this looks like a u, a² + 4, this is the u and this is the 4.1540

There you go, again, this is a general procedure that you can use whenever you have an integral where you have only one factor.1550

Or the rational functional only has one factor and it is an irreducible quadratic.1560

You complete the square on that thing and then you use a u substitution to do what we just did.1564

It will always work.1571

Let me actually write that down.1577

This problem offers a general procedure for dealing with integrals of the type,1581

on the very attractive integral sign, it is b/ ax² + bx + c dx.1611

Anytime you are faced with an integral that looks like that, you can run this procedure.1627

Complete the square and then solve the integral.1631

Now let us deal with our 4th and final case.1637

We just did irreducible quadratic factors that are non-repeating.1640

What if we have repeating quadratic factors?1644

That is actually going to be the same thing.1646

It is just more terms in your partial fraction decomposition.1647

Case 4, it is where our numerator/ our denominator is our rational function.1654

It is where our denominator has repeated irreducible quadratic factors.1664

In other words, it is a quadratic factor of the form ax² + bx + c raised to some power.1686

It itself might be squared, a quadratic factor might show up two times, three times, four times.1713

It is called the algebraic multiplicity, the multiplicity of the factor.1719

The partial fraction decomposition of something like this is as follows.1740

You have a1 x + b1/ ax² + bx + c + a2 x + b2/ ax² + bx + c²,1743

and so on, until you get to a sub n x + b sub n/ ax² + bx + c ⁺n.1770

In other words, whatever n is, you are going to have that many.1783

You are going to have that first power, second power, all the way up to nth power.1789

You are going to have all of these linear factors up on top.1794

You have all of these coefficients to find.1799

Let us do an example, I think it will make sense.1805

Same exact thing that what we did for the repeated linear factors, just have them keep showing up to the nth power.1809

Evaluate 2/ x × x² + 6².1820

In this case, this quadratic factor is irreducible, x² + 6 cannot be factored.1824

It itself is raised to the second power.1829

The denominator is already factored, we do not have to do that.1833

It is already factored, therefore, 2 divided by x, x² + 6².1846

We have a linear factor that is the x, it becomes a/x.1855

We have a quadratic factor raised to the power of 2.1860

We are going to do bx + c/ this quadratic factor to the first power + dx + e, the quadratic factor raised to the second power.1862

Our partial fraction decomposition involves one term where the quadratic factor is to the first power1879

and the second term where the quadratic factor is to the second power.1886

As many terms, all the way up to that many powers.1890

If this were a 3, we would have fx + g/ x² + 1³, and so on.1892

This is it, now we are going to find the least common denominator on this side.1899

Let me actually write this out.1909

The least common denominator here, be very careful, the least common denominator is not this × this × this.1912

It is this × this because this factor is already contained in that.1917

Our lowest common denominator is x × x² + 6², and whatever is on top.1926

Be very careful, you are used to having the least common denominator, just multiply the denominators.1940

Here, because the factors are repeated, this does not need to be this × this × this.1945

It does not need to be that way, this is already contained in that.1951

It is just this and this.1953

Therefore, here what we need to do is we need to do a × x² + 6² + bx + c × x² + 6, only once.1955

And + dx + b × x/ x × x² + 6².1981

This denominator is the same as that denominator, which means the numerator is equivalent to the numerator.2003

Now we expand this numerator and that is what we are going to do next.2009

2 is going to equal, when I multiply all this out.2016

That is fine, it is just algebra.2025

ax⁴ + 12ax² + 36a +, it is going to end up being bx⁴ + cx³ + 6bx² + 6cx + dx² + e ⁺x.2030

When we combine terms, we have an x⁴ term.2070

This is going to be a, it takes care of that one, and a b.2077

It takes care of the x⁴ terms.2084

There is an x³ term.2087

The only x³ term is that one, c.2089

There is an x² term, 12a + 6b + d.2099

Make sure you get all the terms.2113

There is an x term, 6c + e.2116

There is a number term, 36a.2131

All of that is equal to 2.2138

We just set things equal to each other.2143

The equations that we get, in other words, a + b is going to be 0, c is going to be 0.2146

12a + 6b + d is going to be 0.2153

6c + e is going to be 0.2157

36a is going to equal 2, that is what we get.2160

We are going to get a + b = 0.2167

We are going to get c = 0, we are going to get 12a + 6b + d is equal to 0.2172

We are going to get 6c + e is equal to 0.2184

We are going to get 36a is equal to 2.2189

This gives us that a is equal to 1/18.2193

That takes care of that.2198

Let us go to the first one over here.2201

We use the equation a + b is equal to 0, which means that a = -b which means that b = -a which means of b = -1/18.2203

We already know that c is equal to 0.2222

We have taken care of that.2227

Now we have 6c + e is equal to 0.2229

c is equal to 0 so I get 0 + e is equal to 0, which means that e is also equal to 0.2237

Now I’m going to use this equation.2247

12 × 1/18 + 6b, 6 × -1/18 + d is equal to 0.2253

When I solve this, I get d is equal to -6/18.2267

I’m going to leave it as -6/18, instead of reducing that.2273

Therefore, our final partial fraction decomposition, our original function was 2/ x × x² + 6².2276

We said that was equal to a/x + bx + c/ x² + 6 + dx + e/ x² + 6².2289

Now we have a, b, c, d, and e.2304

Our decomposition is 1/18 / x + -1/18 x + c which is 0/ x² + 6 + -6/18 x + e which is 0/ x² + 6².2308

This is our final partial fraction decomposition.2336

That is just the decomposition, that is not the answer.2340

We still have to integrate this thing.2342

The integral of this is the integral of this because these are the same.2346

The integral of this is the integral of this + the integral of that + the integral of that.2351

Nice and simple, the rest is just using all the techniques that we gathered so far.2357

My final answer, our integral is going to equal 1/18 × the integral of 1/x dx - 1/182362

× the integral of x/ x² + 6 dx – 6/18 × the integral of x/ x² + 6² dx.2378

This is equal to 1/18 × natlog of the absolute value of x - 1/18 × ½, because of that u.2403

u = x² + 6, du = 2x dx, x dx = du/2 × natlog of x² + 6 – 6/18 × the integral of x/ x² + 6² dx.2417

How do we handle this?2444

We handle it this way, we do we a u substitution on this.2448

We let u = x² + 6, du = 2x dx, du/2 = x.2452

It is essentially the same thing that we did here, because of that extra x² part,2467

I thought it actually do the u substitution, x dx.2470

We have ½ × the integral of u⁻² du which is equal to ½ × u⁻¹/ -1 which = -1/ 2u which is equal to -1/ 2 × x² + 6.2478

Our final answer is = 1/18 × natlog of x – 1/18 × ½ × natlog of x² + 62509

– 6/18 × -1/ 2 × x² + 6 + c.2531

There we go, partial fraction decompositions are very tedious.2547

They are algebraically intense.2552

There are plenty of places where you can make a mistake but conceptually I do not think it is all together that difficult.2555

You just have to keep track of everything.2560

But this is calculus, you are more than accustomed to that by now because the problems just are,2562

by nature, sort of long and detailed.2569

Thank you so much for joining us here at www.educator.com.2574

We will see you next time, bye.2576

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