Home » Mathematics » College Calculus: Level II » Trigonometric Substitutions

Professor Murray

Professor Murray

Trigonometric Substitutions

Slide Duration:

Table of Contents

Section 1: Advanced Integration Techniques
Integration by Parts

24m 52s

Intro
0:00
Important Equation
0:07
Where It Comes From (Product Rule)
0:20
Why Use It?
0:35
Lecture Example 1
1:24
Lecture Example 2
3:30
Shortcut: Tabular Integration
7:34
Example
7:52
Lecture Example 3
10:00
Mnemonic: LIATE
14:44
Ln, Inverse, Algebra, Trigonometry, e
15:38
Additional Example 4
-1
Additional Example 5
-2
Integration of Trigonometric Functions

25m 30s

Intro
0:00
Important Equation
0:07
Powers (Odd and Even)
0:19
What To Do
1:03
Lecture Example 1
1:37
Lecture Example 2
3:12
Half-Angle Formulas
6:16
Both Powers Even
6:31
Lecture Example 3
7:06
Lecture Example 4
10:59
Additional Example 5
-1
Additional Example 6
-2
Trigonometric Substitutions

30m 9s

Intro
0:00
Important Equations
0:06
How They Work
0:35
Example
1:45
Remember: du and dx
2:50
Lecture Example 1
3:43
Lecture Example 2
10:01
Lecture Example 3
12:04
Additional Example 4
-1
Additional Example 5
-2
Partial Fractions

41m 22s

Intro
0:00
Overview
0:07
Why Use It?
0:18
Lecture Example 1
1:21
Lecture Example 2
6:52
Lecture Example 3
13:28
Additional Example 4
-1
Additional Example 5
-2
Integration Tables

20m

Intro
0:00
Using Tables
0:09
Match Exactly
0:32
Lecture Example 1
1:16
Lecture Example 2
5:28
Lecture Example 3
8:51
Additional Example 4
-1
Additional Example 5
-2
Trapezoidal Rule, Midpoint Rule, Left/Right Endpoint Rule

22m 36s

Intro
0:00
Trapezoidal Rule
0:13
Graphical Representation
0:20
How They Work
1:08
Formula
1:47
Why a Trapezoid?
2:53
Lecture Example 1
5:10
Midpoint Rule
8:23
Why Midpoints?
8:56
Formula
9:37
Lecture Example 2
11:22
Left/Right Endpoint Rule
13:54
Left Endpoint
14:08
Right Endpoint
14:39
Lecture Example 3
15:32
Additional Example 4
-1
Additional Example 5
-2
Simpson's Rule

21m 8s

Intro
0:00
Important Equation
0:03
Estimating Area
0:28
Difference from Previous Methods
0:50
General Principle
1:09
Lecture Example 1
3:49
Lecture Example 2
6:32
Lecture Example 3
9:07
Additional Example 4
-1
Additional Example 5
-2
Improper Integration

44m 18s

Intro
0:00
Horizontal and Vertical Asymptotes
0:04
Example: Horizontal
0:16
Formal Notation
0:37
Example: Vertical
1:58
Formal Notation
2:29
Lecture Example 1
5:01
Lecture Example 2
7:41
Lecture Example 3
11:32
Lecture Example 4
15:49
Formulas to Remember
18:26
Improper Integrals
18:36
Lecture Example 5
21:34
Lecture Example 6 (Hidden Discontinuities)
26:51
Additional Example 7
-1
Additional Example 8
-2
Section 2: Applications of Integrals, part 2
Arclength

23m 20s

Intro
0:00
Important Equation
0:04
Why It Works
0:49
Common Mistake
1:21
Lecture Example 1
2:14
Lecture Example 2
6:26
Lecture Example 3
10:49
Additional Example 4
-1
Additional Example 5
-2
Surface Area of Revolution

28m 53s

Intro
0:00
Important Equation
0:05
Surface Area
0:38
Relation to Arclength
1:11
Lecture Example 1
1:46
Lecture Example 2
4:29
Lecture Example 3
9:34
Additional Example 4
-1
Additional Example 5
-2
Hydrostatic Pressure

24m 37s

Intro
0:00
Important Equation
0:09
Main Idea
0:12
Different Forces
0:45
Weight Density Constant
1:10
Variables (Depth and Width)
2:21
Lecture Example 1
3:28
Additional Example 2
-1
Additional Example 3
-2
Center of Mass

25m 39s

Intro
0:00
Important Equation
0:07
Main Idea
0:25
Centroid
1:00
Area
1:28
Lecture Example 1
1:44
Lecture Example 2
6:13
Lecture Example 3
10:04
Additional Example 4
-1
Additional Example 5
-2
Section 3: Parametric Functions
Parametric Curves

22m 26s

Intro
0:00
Important Equations
0:05
Slope of Tangent Line
0:30
Arc length
1:03
Lecture Example 1
1:40
Lecture Example 2
4:23
Lecture Example 3
8:38
Additional Example 4
-1
Additional Example 5
-2
Polar Coordinates

30m 59s

Intro
0:00
Important Equations
0:05
Polar Coordinates in Calculus
0:42
Area
0:58
Arc length
1:41
Lecture Example 1
2:14
Lecture Example 2
4:12
Lecture Example 3
10:06
Additional Example 4
-1
Additional Example 5
-2
Section 4: Sequences and Series
Sequences

31m 13s

Intro
0:00
Definition and Theorem
0:05
Monotonically Increasing
0:25
Monotonically Decreasing
0:40
Monotonic
0:48
Bounded
1:00
Theorem
1:11
Lecture Example 1
1:31
Lecture Example 2
11:06
Lecture Example 3
14:03
Additional Example 4
-1
Additional Example 5
-2
Series

31m 46s

Intro
0:00
Important Definitions
0:05
Sigma Notation
0:13
Sequence of Partial Sums
0:30
Converging to a Limit
1:49
Diverging to Infinite
2:20
Geometric Series
2:40
Common Ratio
2:47
Sum of a Geometric Series
3:09
Test for Divergence
5:11
Not for Convergence
6:06
Lecture Example 1
8:32
Lecture Example 2
10:25
Lecture Example 3
16:26
Additional Example 4
-1
Additional Example 5
-2
Integral Test

23m 26s

Intro
0:00
Important Theorem and Definition
0:05
Three Conditions
0:25
Converging and Diverging
0:51
P-Series
1:11
Lecture Example 1
2:19
Lecture Example 2
5:08
Lecture Example 3
6:38
Additional Example 4
-1
Additional Example 5
-2
Comparison Test

22m 44s

Intro
0:00
Important Tests
0:01
Comparison Test
0:22
Limit Comparison Test
1:05
Lecture Example 1
1:44
Lecture Example 2
3:52
Lecture Example 3
6:01
Lecture Example 4
10:04
Additional Example 5
-1
Additional Example 6
-2
Alternating Series

25m 26s

Intro
0:00
Main Theorems
0:05
Alternation Series Test (Leibniz)
0:11
How It Works
0:26
Two Conditions
0:46
Never Use for Divergence
1:12
Estimates of Sums
1:50
Lecture Example 1
3:19
Lecture Example 2
4:46
Lecture Example 3
6:28
Additional Example 4
-1
Additional Example 5
-2
Ratio Test and Root Test

33m 27s

Intro
0:00
Theorems and Definitions
0:06
Two Common Questions
0:17
Absolutely Convergent
0:45
Conditionally Convergent
1:18
Divergent
1:51
Missing Case
2:02
Ratio Test
3:07
Root Test
4:45
Lecture Example 1
5:46
Lecture Example 2
9:23
Lecture Example 3
13:13
Additional Example 4
-1
Additional Example 5
-2
Power Series

38m 36s

Intro
0:00
Main Definitions and Pattern
0:07
What Is The Point
0:22
Radius of Convergence Pattern
0:45
Interval of Convergence
2:42
Lecture Example 1
3:24
Lecture Example 2
10:55
Lecture Example 3
14:44
Additional Example 4
-1
Additional Example 5
-2
Section 5: Taylor and Maclaurin Series
Taylor Series and Maclaurin Series

30m 18s

Intro
0:00
Taylor and Maclaurin Series
0:08
Taylor Series
0:12
Maclaurin Series
0:59
Taylor Polynomial
1:20
Lecture Example 1
2:35
Lecture Example 2
6:51
Lecture Example 3
11:38
Lecture Example 4
17:29
Additional Example 5
-1
Additional Example 6
-2
Taylor Polynomial Applications

50m 50s

Intro
0:00
Main Formulas
0:06
Alternating Series Error Bound
0:28
Taylor's Remainder Theorem
1:18
Lecture Example 1
3:09
Lecture Example 2
9:08
Lecture Example 3
17:35
Additional Example 4
-1
Additional Example 5
-2
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