Integrals of Natural Logarithmic Functions, Calculus AB

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Integrals of Natural Logarithmic Functions

Try to transform expression into a fraction with only a constant in the numerator
Substitute denominator with u

All other integration rules still apply

Integrals of Natural Logarithmic Functions

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

Types of Integrals: Natural Log Functions

Example 1

Example 2

Example 3

Example 4

Example 5

Example 6

I. Functions
	Definitions & Properties of Functions	11:26
	Graphing	13:58
	Inverse Functions	6:47
	Polynomial Functions	5:04
	Trigonometric Functions	6:45
	Inverse Trigonometric Functions	5:58
	Trigonometric Identities	17:42
	Exponential Functions	5:53
	Logarithmic Functions	7:08
	Rational Functions	15:36
	Conic Sections	14:58
II. Limits and Continuity
	Limit Definition & Properties	7:15
	Solving Limits with Algebra	8:01
	Rational Limit Rules	3:16
	One Sided Limits	9:57
	Special Trigonometric Limits	8:28
	Limits & Continuity	10:14
	Limits: Multiple Choice Practice	6:16
III. Derivatives
	Derivative Definition & Properties	4:11
	Basic Rules of Differentiation	7:07
	Power Rule	7:14
	Trigonometric Rules	7:53
	Product Rule	11:11
	Quotient Rule	16:50
	Chain Rule	19:48
	Higher Order Derivatives	15:00
	Derivatives of Exponential Functions	13:14
	Derivatives of Logarithmic Functions	11:30
	Derivatives of Inverse Trigonometric Functions	16:54
	Implicit Differentiation	16:53
	Multiple Choice Practice: Derivatives	11:07
IV. Applications of Derivatives
	Tangent & Normal Lines	22:36
	Position Velocity & Acceleration	18:42
	Related Rates	26:22
	Minimum & Maximum	12:22
	Concavity	11:43
	Rolles Theorem	8:28
	Mean Value Theorem	9:39
	Differentials	12:25
	Applications of Derivatives: Multiple Choice Practice	13:21
	Applications of Derivatives: Free Response Practice	10:22
V. Integrals
	Definition of Integrals	1:08
	Integrals of Power Rule	8:50
	Integrals Basic Rules of Integration	9:43
	Trigonometric Rules of Integrals	8:58
	Chain Rule	13:59
	Integrals of Exponential Functions	12:52
	Integrals of Natural Logarithmic Functions	13:00
	Integrals of Inverse Trigonometric Functions	8:29
	Integrals: Multiple Choice Practice	15:37
VI. Applications of Integrals
	Fundamental Theorem of Calculus	15:55
	Area Under A Curve	18:34
	Reimann Sums	10:35
	Trapezoid Rule	12:46
	Mean Value Theorem	11:22
	Second Fundamental Theorem of Calculus	4:44
	Area Between Curves	16:39
	Revolving Solids Washer Disk Methods	21:09
	Revolving Solids Cylindrical Shells Method	26:46
	Revolving Solids Known Cross Sections	27:41
	Differential Equations Eulers Method	17:54
	Differential Equations Slope Fields	16:30
	Application of Integrals: Multiple Choice Practice	14:19
	Application of Integrals: Free Response Practice	9:14
VII. Sample AP Test
	AP Calculus AB Practice test: Section 1: Multiple Choice Part 1	17:50
	AP Calculus AB Practice test: Section 1: Multiple Choice Part 2	17:32
	AP Calculus AB Practice test: Section 1: Multiple Choice Part 3	22:14
	AP Calculus AB Practice test: Section 1: Multiple Choice Part 4	19:35
	AP Calculus AB Practice test: Section 1: Multiple Choice Part 5	25:43
	AP Calculus AB Practice Test: Section 2: Free Response Part 1	16:50
	AP Calculus AB Practice Test: Section 2: Free Response Part 2	21:36
	AP Calculus AB Practice Test: Section 2: Free Response Part 3	13:39

Mathematics: AP Calculus AB

I. Functions
	Definitions & Properties of Functions			11:26
		Intro		0:00
		Definition		0:28
		Properties: Vertical Line Test		1:32
			Domain	1:38
			Range	1:59
			Vertical Line test	2:19
			Example 1	2:33
			Example 2	3:10
		Properties: Roots or Zeros		4:04
			Finding the Root	4:16
		Properties: Forms		5:12
			Graphically	5:20
			List	5:46
			Equation	6:11
			Function	6:38
		Properties: Odd & Even		7:12
			Even Function	7:14
			Odd Function	8:25
		Properties: Increasing & Decreasing		9:17
			Increasing Function	9:22
			Decreasing Function	10:21
	Graphing			13:58
		Intro		0:00
		Manipulating		0:10
			A in the Equation	0:39
			B in the Equation	0:44
			C & D in the Equation	0:49
			Negative values	0:59
			Example 1	1:17
		Example 2		1:51
		Example 3: Absolute Value Functions		3:43
			Example 4	4:57
		Example 5		6:17
		Example 6		8:02
		Example 7		9:10
		Example 8		11:02
		Example 9		11:47
	Inverse Functions			6:47
		Intro		0:00
		Inverse		0:08
			Definition	0:18
			Example: Finding the Inverse	1:03
		Example 2		2:29
		Example 3		3:12
		Example 4		4:41
	Polynomial Functions			5:04
		Intro		0:00
		Types of Functions: Polynomials		0:07
			No Domain Restrictions	0:12
			No Discontinuities	0:19
			Degree Test	0:31
		Types of Functions: Polynomials		1:17
			Leading Coefficient Test	1:33
			Leading Coefficient Positive, Even Degree	1:54
			Leading Coefficient Positive, Odd Degree	2:13
			Leading Coefficient Negative Even Degree	2:34
			Leading Coefficient Positive, Odd Degree	2:46
		Examples: Types of Functions: Polynomials		3:03
		Examples: Types of Functions: Polynomials		4:18
	Trigonometric Functions			6:45
		Intro		0:00
		Types of Functions: Trigonometric		0:05
			6 Functions To Be Familiar With	0:14
		Example 1: SIN		1:38
		Example 2: COS		3:22
		Example 3: TAN		4:38
	Inverse Trigonometric Functions			5:58
		Intro		0:00
		Types of Functions: Trigonometric- Inverse Trig Functions		0:07
		Example: Inverse SIN of X		0:45
		Example: Inverse Function		2:30
		Example: Inverse TAN of X		4:42
	Trigonometric Identities			17:42
		Intro		0:00
		Types of Functions: Trigonometric- Trig Identities		0:07
			4 Identities	0:24
			Pythagorean	0:28
			Double Angle	1:10
			Power Reducing	1:28
			Sum or Difference	1:42
		Couple More Identities		1:59
			Negative Angle	2:04
			Product to Sum	2:39
		Example 1: Prove		3:00
		Example 2: Simplify Expression		5:02
		Example 3: Prove		5:56
		Example 4: Prove		8:02
		Example 5: Prove With TAN		12:43
	Exponential Functions			5:53
		Intro		0:00
		Types of Functions: Exponentials		0:07
			General Form	0:10
			Special Exponential Function	0:17
		Example 1: Using Exponential Properties		0:46
		Example 2: Using Exponential Properties		1:58
		Example 3: Using Trig Identities & Exponential Properties		3:16
		Example 4: Using Exponential Properties		4:37
	Logarithmic Functions			7:08
		Intro		0:00
		Types of Functions: Logarithmic		0:06
			General Form	0:10
			2 Special Logarithmic Func.	0:19
			Euler's # / Natural Log	0:27
		Logarithmic & Exponential Relationship		0:45
			Log form	1:56
		Properties		2:09
		Example 1: Apply Basic Principle of Log Func.		3:05
		Example 2: Use Properties		3:40
		Example 3: Regular Log		5:16
	Rational Functions			15:36
		Intro		0:00
		Types of Functions: Rational - Definition		0:06
		Example 1: Graph Rational Func.		0:36
		Example 2: Find Asymptotes of Func.		7:02
		Example 3: Find Asymptotes of Func.		8:59
		Example 4: Graph Rational Func.		11:08
	Conic Sections			14:58
		Intro		0:00
		Types of Conic Sections		0:06
			Parabolas	0:19
			Circles	1:36
			Ellipses	2:40
			Hyperbolas	4:42
		Complete the Square		6:40
		Example: Conic Sections		9:08
		Example 2: Conic Sections		10:59
		Example 3: Graph Conic Sections		12:21
II. Limits and Continuity
	Limit Definition & Properties			7:15
		Intro		0:00
		Definition		0:06
			Example: Limit	0:17
		Properties		1:13
			1st Property	1:21
			2nd Property	1:34
			Special Property	1:51
		Limits		2:36
			Explain Example	2:49
		Limits Example		4:39
		Limits Example		5:21
	Solving Limits with Algebra			8:01
		Intro		0:00
		Solving Limits with Algebra		0:07
			Example 1: Solve Algebraically	0:30
		Solving Limits with Algebra, Example 2		2:28
		Solving Limits with Algebra, Example 3		3:18
		Solving Limits with Algebra, Example 4		4:56
		Solving Limits with Algebra, Example 5		6:26
	Rational Limit Rules			3:16
		Intro		0:00
		Rational Limit Rules		0:07
			Review of Solving Problem Algebraically	0:08
			Limit Rules	0:28
			Rule 1	0:35
			Rule 2	0:40
			Rule 3	0:45
		Rational Limit Rules		1:02
			Applying 1st Rule	1:22
		Rational Limit Rules		1:50
			Applying 2nd Rule	2:09
		Rational Limit Rules		2:26
			Applying 3rd Rule	2:40
	One Sided Limits			9:57
		Intro		0:00
		Types of Limits: One-Sided Limit Rules		0:06
			Example	0:19
			Applying Same Rule	0:34
			Rule to Keep In Mind	0:52
		Types of Limits: One-Sided Limit Example 1		1:12
			Limit of x² From Negative Side	2:11
		Types of Limits: One-Sided Limit, Example 2		2:27
		Types of Limits: One-Sided Limit, Example 3		4:26
		Types of Limits: One-Sided Limit, Example 4		5:47
		One-Sided Limit Example: X with Even Degree Polynomial		7:00
		One-Sided Limit Example: Entire Denominator Squared		8:09
	Special Trigonometric Limits			8:28
		Intro		0:00
		Types of Limits: Special Trig Limits		0:07
			Pre-set Rules	0:35
		Special Trig Limits, Example 1		0:58
		Special Trig Limits, Example 2		2:50
		Special Trig Limits, Example 3		3:55
		Special Trig Limits, Example 4: With More Degrees		4:57
		Special Trig Limits, Example 5		6:21
	Limits & Continuity			10:14
		Intro		0:00
		Definition		0:06
			3 Rules: f(x) Is Continuous…	0:21
		Example 1: Finding Continuity		1:06
		Types of Discontinuity		2:44
			Jump	2:52
			Point	3:24
			Essential (Asymptote)	3:47
			Removable	4:17
		Example 2: Continuity Examples		4:41
		Example 3: Continuity Examples		6:13
		Example 4: Locate & Identify Type of Discontinuities		8:00
	Limits: Multiple Choice Practice			6:16
		Intro		0:00
		Problem 1		0:08
		Problem 2		1:51
		Problem 3		2:54
		Problem 4		4:31
III. Derivatives
	Derivative Definition & Properties			4:11
		Intro		0:00
		Definition		0:09
			Formal Definition	0:45
			Difference Quotient	1:12
		Basic Derivatives		1:16
		Differentiability		2:54
	Basic Rules of Differentiation			7:07
		Intro		0:00
		Basic Rules of Differentiation		0:09
		Constant Rule		0:14
		Constant Multiple Rule		1:10
		Addition and Difference Rule		1:40
		Example 1: Constant Rule		2:25
		Example 2: Constant Multiple Rule		3:01
		Example 3: Constant Multiple Rule		3:35
		Example 4: Constant Rule		4:34
		Example 5: Constant Multiple Rule		5:03
		Example 6		5:33
	Power Rule			7:14
		Intro		0:00
		Power Rule		0:07
			Power Rule Definition	0:30
		Example 1		1:11
		Example 2		2:25
		Example 3		3:05
		Example 4		4:18
		Example 5		5:13
	Trigonometric Rules			7:53
		Intro		0:00
		Trigonometric Rules		0:07
			COS X	0:38
			Find Derivative	1:02
		Example 1		2:46
		Example 2: COS Function		3:09
		Example 3: Composite Expression		3:54
		Example 4: Sec Function		5:02
		Example 5: CSC		5:33
		Example 6L COT		6:42
	Product Rule			11:11
		Intro		0:00
		Product Rule		0:07
			Definition	0:20
			Example 1	0:43
		Example 2		2:11
		Example 3		4:24
		Example 4		5:24
		Example 5		6:42
		Example 6		7:51
	Quotient Rule			16:50
		Intro		0:00
		Quotient Rule		0:07
			Definition	0:30
			Example 1	1:17
		Example 2: With No X In Numerator		2:49
		Example 3		4:30
		Example 4: With Decimals		6:46
		Example 5		8:53
		Example 6: With Trig Functions		12:55
	Chain Rule			19:48
		Intro		0:00
		Chain Rule		0:07
			Definition	0:17
			Example 1: Applying the Chain Rule	1:33
		Example 2		4:25
		Example 3		6:02
		Example 4		9:25
		Example 5		12:47
		Example 6		15:27
	Higher Order Derivatives			15:00
		Intro		0:00
		Types of Derivatives: Higher Order Derivatives		0:07
			1st Derivative / F Prime	0:19
			2nd Derivative	0:25
			3rd Derivative	0:32
			Example 1	1:48
		Example 2: Find 3rd Derivative		3:13
		Example 3: Acceleration		4:25
		Example 4		10:20
		Example 5: 2nd Derivative		12:11
	Derivatives of Exponential Functions			13:14
		Intro		0:00
		Types of Derivatives: Exponential Functions		0:08
			Derivatives: Definition/ Formula	0:28
			Example 1	1:25
		Example 2		2:47
		Example 3		4:13
		Example 4		7:11
		Example 5		9:23
		Example 6		11:06
	Derivatives of Logarithmic Functions			11:30
		Intro		0:00
		Types of Derivatives: Logarithmic Functions		0:06
			Rule for Logarithmic Functions	0:28
			Example 1	0:58
		Example 2		3:10
		Example 3		4:38
		Example 4		7:18
		Example 5		8:48
		Example 6		9:38
	Derivatives of Inverse Trigonometric Functions			16:54
		Intro		0:00
		Types of Derivatives: Inverse Trigonometric Functions		0:06
			6 Fundamental Properties of Inverse Trigonometric Functions	0:38
		Example 1		2:17
		Example 2		3:41
		Example 3		5:37
		Example 4		7:24
		Example 5		10:08
	Implicit Differentiation			16:53
		Intro		0:00
		Implicit Differentiation: First Order		0:07
			Example 1: Setting Up	0:45
			Example 1: Solving	1:41
		Implicit Differentiation: Second Order (Ex. 2)		4:55
		Example 3: Implicit Differentiation		9:11
		Example 4: Implicit Differentiation		9:56
		Example 5: Implicit Differentiation With Double Derivative		12:46
	Multiple Choice Practice: Derivatives			11:07
		Intro		0:00
		Practice Problem 1		0:09
			Answer	3:24
		Practice Problem 2		3:36
			Answer	6:29
		Practice Problem 3		6:42
			Answer	8:39
		Practice Problem 4		8:43
			Answer	9:33
		Practice Problem 5		9:41
			Answer	10:40
IV. Applications of Derivatives
	Tangent & Normal Lines			22:36
		Intro		0:00
		Tangent and Normal Lines		0:10
			Definition	0:22
			Example 1	0:55
		Tangent and Normal Lines: Example 2		2:43
		Tangent and Normal Lines		5:21
			Example 3	5:35
		Tangent and Normal Lines: Example 4		9:14
		Tangent and Normal Lines: Example 5		12:27
		Tangent and Normal Lines: Example 6		15:54
		Tangent and Normal Lines: Example 7		19:05
	Position Velocity & Acceleration			18:42
		Intro		0:00
		Position, Velocity, and Acceleration		0:10
			Position Function	0:14
			Velocity Function	0:34
			Acceleration Function	1:01
		Example 1		1:20
		Example 2		6:31
		Example Continue: Velocity When Acceleration is Zero		6:32
		Example 3: Where Is Particle Changing Directions?		8:16
		Example 4: Total Distance Traveled From 0 to 2 Second		11:09
		Example 5: Ball Drop Problem		16:40
	Related Rates			26:22
		Intro		0:00
		Related Rates		0:06
			Finding Rate of Change: Organization & Big Picture	0:23
		Example 2: Area of a Circle		1:17
		Example 3: Spherical Volume Expanding		4:19
		Example 4: Traveling Problem		7:57
		Example 5: Square Increase		12:37
		Example 6: Standard Related Rates Problem		16:59
		Example 7: Standard Related Rates Problem		19:49
	Minimum & Maximum			12:22
		Intro		0:00
		Extrema: First Derivative Test		0:09
			Example 1	0:46
		Example 2: Real World Application/ Cost Function		4:05
		Example 3: Minimums & Maximums		7:10
		Example 4: Find Critical Points		10:52
	Concavity			11:43
		Intro		0:00
		Concavity: Second Derivative Test		0:06
			Definition	0:34
			Example 1	0:54
		Example 2		2:51
		Example 3		4:08
		Example 4		5:52
	Rolles Theorem			8:28
		Intro		0:00
		Rolle's Theorem		0:07
			Conditions	0:11
			Summary	0:41
		Example 1		1:09
		Example 2		3:08
		Example 3		4:48
	Mean Value Theorem			9:39
		Intro		0:00
		Mean Value Theorem		0:06
			Rolle's Theorem	0:07
			Mean Value Theorem Conditions	0:24
			Mean Value Theorem Definition	0:36
		Example 1		0:56
		Example 2		2:44
		Example 3		5:28
		Example 4		7:15
	Differentials			12:25
		Intro		0:00
		Differentials		0:08
			1st Differential Formula	0:29
			2nd Differential Formula	0:57
		Example 1		1:06
		Example 2		3:21
		Example 3		5:49
		Example 4		7:19
		Example 5		9:06
	Applications of Derivatives: Multiple Choice Practice			13:21
		Intro		0:00
		Practice Problem 1		0:10
			Answer	1:57
		Practice Problem 2		2:08
			Answer	5:39
		Practice Problem 3		5:45
			Answer	9:59
		Practice Problem 4		10:12
			Answer	11:49
		Practice Problem 5		11:52
			Answer	13:00
	Applications of Derivatives: Free Response Practice			10:22
		Intro		0:00
		Practice Problem 1		0:10
			Slope	1:30
			Tangent Line Equation	2:17
		Absolute Minimum		2:24
			2 Possible X Points With Minimums	3:15
			One Interest Point	4:14
			Concavity	4:33
			Positive Value = Positive Concavity	4:10
			Minimum Point	5:34
			Absolute Minimum	6:18
		Point(s) of Inflection		6:31
			Definition	6:49
			2 Points Of Inflection	9:59
V. Integrals
	Definition of Integrals			1:08
		Intro		0:00
		Definition		0:09
			Definition	0:16
			Example	0:20
	Integrals of Power Rule			8:50
		Intro		0:00
		Power Rule		0:06
			Example 1	0:25
		Example 2		2:02
		Example 3		2:54
		Example 4		3:45
		Example 5		4:49
		Example 6		6:47
	Integrals Basic Rules of Integration			9:43
		Intro		0:00
		Basic Rules of Integration		0:09
			Constant Rule	0:22
			Example 1	0:40
		Addition and Difference Rule		1:40
			Example 2	1:58
		Example 3: Subtraction/ Difference Rule		2:47
		Example 4		3:55
		Example 5		5:19
		Example 6		7:37
	Trigonometric Rules of Integrals			8:58
		Intro		0:00
		Trigonometric Rules		0:09
			Integral of SIN	0:38
		Example 1: Integral of SIN		1:46
		Example 2: Integral of COS		2:38
		Example 3: With 2 terms of X		3:06
		Example 4: Integral of SEC		4:15
		Example 5: Integral of CSC		5:06
		Example 6		6:18
	Chain Rule			13:59
		Intro		0:00
		Chain Rule		0:07
			Example 1	0:37
		Example 2		3:17
		Example 3		5:09
		Example 4		7:53
		Example 5		9:40
		Example 6		11:39
	Integrals of Exponential Functions			12:52
		Intro		0:00
		Types of Integrals: Exponential Functions		0:09
			Rule 1	0:30
			Rule 2	0:49
			Example 1	1:11
		Example 2		2:54
		Example 3		4:19
		Example 4		5:19
		Example 5		7:37
		Example 6		9:04
	Integrals of Natural Logarithmic Functions			13:00
		Intro		0:00
		Types of Integrals: Natural Log Functions		0:09
			Example 1	0:49
		Example 2		2:06
		Example 3		4:01
		Example 4		5:37
		Example 5		7:30
		Example 6		9:05
	Integrals of Inverse Trigonometric Functions			8:29
		Intro		0:00
		Types of Integrals: Inverse Trig Functions		0:09
			One Property	0:40
		Example 1		1:19
		Example 2		3:44
		Example 3		4:53
		Example 4		5:53
	Integrals: Multiple Choice Practice			15:37
		Intro		0:00
		Problem 1		0:09
			Answer	4:09
		Problem 2		4:33
			Answer	5:54
		Problem 3		5:59
			Answer	8:02
		Problem 4		8:06
			Answer	10:27
		Problem 5		10:43
			Answer	14:46
VI. Applications of Integrals
	Fundamental Theorem of Calculus			15:55
		Intro		0:00
		Fundamental Theorem of Calculus: Properties		0:10
			Definition of Integral	0:49
			Example 1	1:14
		Fundamental Theorem of Calculus: Properties		2:40
			Rule 1	2:50
			Rule 2	3:14
			Rule 3	3:33
			Rule 4	3:52
		Example 2		4:07
		Example 3		6:17
		Example 4		9:31
		Example 5		10:52
		Example 6		13:34
	Area Under A Curve			18:34
		Intro		0:00
		Area Under Curve		0:07
			Definition of Integral	0:09
			Left Endpoint	1:17
			Right Endpoint	1:47
			Midpoints	2:09
		Example 1		2:40
		Example 2		4:59
		Example 3		8:48
		Example 4		10:23
		Example 5		12:30
		Example 6		15:32
	Reimann Sums			10:35
		Intro		0:00
		Reimann Sums		0:08
			Definition	1:07
		Example 1		2:48
		Example 2		5:38
		Example 3		7:21
		Example 4		9:14
	Trapezoid Rule			12:46
		Intro		0:00
		The Trapezoid Rule		0:09
			Definition: Area Of A Trapezoid	0:26
			Terms of Formula	1:35
		Example 1		2:11
		Example 2		4:29
		Example 3		7:22
		Example 4		10:01
	Mean Value Theorem			11:22
		Intro		0:00
		Mean Value Theorem of Integration		0:06
			Example 1	0:53
		Example 2		2:29
		Example 3		3:48
		Example 4		6:02
	Second Fundamental Theorem of Calculus			4:44
		Intro		0:00
		Second Fundamental Theorem of Calculus		0:07
			Definition	0:39
		Example 1		1:08
		Example 2		2:07
		Example 3		2:48
		Example 4		3:23
	Area Between Curves			16:39
		Intro		0:00
		Example 1		0:10
		Example 2		3:00
		Example 3		4:46
		Example 4		8:22
		Example 5		11:04
		Example 6		13:09
	Revolving Solids Washer Disk Methods			21:09
		Intro		0:00
		Revolving Solids Washer Disk Methods		0:11
			Explanation	0:33
			Formula	3:12
		Example 1		3:42
		Example 2		6:54
		Example 3		9:29
		Example 4		12:16
		Example 5		15:35
	Revolving Solids Cylindrical Shells Method			26:46
		Intro		0:00
		Revolving Solids: Cylindrical Shells Method		0:09
			Volume Of A Solid	0:25
			Formula	2:51
		Example 1		2:56
		Example 2		7:28
		Example 3		11:39
		Example 4		17:36
		Example 5		21:45
	Revolving Solids Known Cross Sections			27:41
		Intro		0:00
		Revolving Solids Known Cross Sections		0:08
			Example 1	0:35
		Example 2		6:01
		Example 3		11:03
		Example 4		17:29
		Example 5		22:19
	Differential Equations Eulers Method			17:54
		Intro		0:00
		Differential Equations		0:08
			Example 1	0:30
		Differential Equations: Euler's Method		2:33
			Rules	2:39
			Example 2	3:00
		Example 3		5:42
		Example 4		9:44
		Example 5		14:14
	Differential Equations Slope Fields			16:30
		Intro		0:00
		Slope Fields		0:08
			What Are Slope Fields	0:21
			Example 1	0:42
		Example 2		6:30
		Example 3		11:17
	Application of Integrals: Multiple Choice Practice			14:19
		Intro		0:00
		Practice Problem 1		0:10
			Answer	3:46
		Practice Problem 2		3:49
			Answer	6:20
		Practice Problem 3		6:26
			Answer	8:02
		Practice Problem 4		8:07
			Answer	10:58
		Practice Problem 5		11:05
			Answer	14:06
	Application of Integrals: Free Response Practice			9:14
		Intro		0:00
		Problem 1		0:10
			Part A	0:24
			Part A: Solution	2:04
			Part B	2:10
		Problem 1, Part B Continue		2:23
			Part B: Solution	6:15
		Problem 1, Part C		6:58
			Part C: Solution	12:40
		Problem 2		12:52
			Part A	13:02
			Part A: Solution	15:34
			Part B	16:03
			Part B: Solution	18:48
VII. Sample AP Test
	AP Calculus AB Practice test: Section 1: Multiple Choice Part 1			17:50
		Intro		0:00
		Problem 1		0:20
		Problem 2		1:24
		Problem 3		2:53
		Problem 4		3:56
		Problem 5		8:18
		Problem 6		9:06
		Problem 7		10:14
		Problem 8		12:16
		Problem 9		14:13
	AP Calculus AB Practice test: Section 1: Multiple Choice Part 2			17:32
		Intro		0:00
		Problem 10		0:18
		Problem 11		2:26
		Problem 12		6:11
		Problem 13		7:04
		Problem 14		8:06
		Problem 15		10:32
		Problem 16		11:40
		Problem 17		13:00
		Problem 18		14:43
	AP Calculus AB Practice test: Section 1: Multiple Choice Part 3			22:14
		Intro		0:00
		Problem 19		0:21
		Problem 20		2:33
		Problem 21		7:23
		Problem 22		10:24
		Problem 23		12:18
		Problem 24		13:13
		Problem 25		15:52
		Problem 26		17:03
		Problem 27		19:44
	AP Calculus AB Practice test: Section 1: Multiple Choice Part 4			19:35
		Intro		0:00
		Problem 28		0:23
		Problem 29		3:50
		Problem 30		5:31
		Problem 31		9:02
		Problem 32		10:07
		Problem 33		11:27
		Problem 34		13:47
		Problem 35		15:21
		Problem 36		16:53
	AP Calculus AB Practice test: Section 1: Multiple Choice Part 5			25:43
		Intro		0:00
		Problem 37		0:22
		Problem 38		2:27
		Problem 39		5:36
		Problem 40		7:21
		Problem 41		10:08
		Problem 42		11:29
		Problem 43		13:07
		Problem 44		18:18
		Problem 45		21:08
	AP Calculus AB Practice Test: Section 2: Free Response Part 1			16:50
		Intro		0:00
		Problem 1, Part A		0:20
		Problem 1, Part B		3:03
		Problem 1, Part C		4:11
		Problem 1, Part D		5:36
		Problem 2, Part A		7:37
		Problem 2, Part B		9:02
		Problem 2, Part C		12:31
	AP Calculus AB Practice Test: Section 2: Free Response Part 2			21:36
		Intro		0:00
		Problem 3, Part A		0:18
		Problem 3, Part B		5:57
		Problem 4, Part A		11:26
		Problem 4, Part B		12:28
		Problem 4, Part C		15:35
		Problem 4, Part D		18:56
	AP Calculus AB Practice Test: Section 2: Free Response Part 3			13:39
		Intro		0:00
		Problem 5, Part A		0:21
		Problem 5, Part B		3:07
		Problem 5, Part C		6:43
		Problem 6		8:24

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Integrals of Natural Logarithmic Functions

Integrals of Natural Logarithmic Functions

Mathematics: AP Calculus AB

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