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Calculus AB Integrals of Inverse Trigonometric Functions

Section 5: Integrals: Lecture 8 | 8:29 min

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Lecture Comments (7)

	0 answers Post by Arjun Chaturvedi on April 25, 2014 Can't see the rest of video.
	0 answers Post by Akilah Miller on February 10, 2014 I do not believe, for example 3, that the question involves the inverse trig identities on the account that it does not match any of them. So, I used u-substitution instead and the result was 4/3(x^2)^3/4 + C. Is this correct or am I missing something?
	1 answer Last reply by: Arjun Chaturvedi Fri Apr 25, 2014 8:54 PM Post by sergey smith on October 29, 2013 wouldn't #3 be 2arcsinh(x)+C
	0 answers Post by Latricia Glavin on June 11, 2013 uhm I believe the correct answer is 4arctanx + c to example 3
	1 answer Last reply by: Colby Trace Thu Apr 4, 2013 6:02 PM Post by Shehan Gunasekara on June 11, 2012 Example 3 was great ;P

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Integrals of Inverse Trigonometric Functions

Memorize!
Manipulate expression to resemble that of the inverse trig formula to apply

Integrals of Inverse Trigonometric Functions

∫[1/(√{x² − √2})] dx

∫[1/(√{x² − √2})] dx = ∫[1/(√{x² − (2^[1/4])²})] dx

∫[1/(√{x² − √2})] dx = sin⁻¹ [x/(2^[1/4])] + c

∫[cosx/(sin² x + 4)] dx

u = sinx
du = cosx dx
∫[cosx/(sin² x + 4)] dx = ∫[1/(u² + 4)] du
∫[cosx/(sin² x + 4)] dx = ∫[1/(2² + u²)] du
∫[cosx/(sin² x + 4)] dx = [1/2] tan⁻¹ [u/2] + c

∫[cosx/(sin² x + 4)] dx = [1/2] tan⁻¹ ([sinx/2]) + c

∫[(e^x)/(√{e^4x − e^2x})] dx

∫[(e^x)/(√{e^4x − e^2x})] dx = ∫[(e^x)/(√{(e^2x)(e^2x − 1)})] dx
∫[(e^x)/(√{e^4x − e^2x})] dx = ∫[(e^x)/(e^x √{e^2x − 1})] dx
∫[(e^x)/(√{e^4x − e^2x})] dx = ∫[(e^x)/(e^x √{(e^x)² − 1²})] dx
u = e^x
du = e^x dx
∫[(e^x)/(√{e^4x − e^2x})] dx = ∫[1/(u √{u² − 1²})] du
∫[(e^x)/(√{e^4x − e^2x})] dx = [1/1] sec⁻¹ [u/1] + c

∫[(e^x)/(√{e^4x − e^2x})] dx = sec⁻¹ e^x + c

∫[1/(√{−x² − 2x})] dx

∫[1/(√{−x² − 2x})] dx = ∫[1/(√{1 − (x² + 2x + 1)})] dx
∫[1/(√{−x² − 2x})] dx = ∫[1/(√{1 − (x + 1)²})] dx
u = x + 1
du = dx
∫[1/(√{−x² − 2x})] dx = ∫[1/(√{1 − u²})] du
∫[1/(√{−x² − 2x})] dx = sin⁻¹ u + c

∫[1/(√{−x² − 2x})] dx = sin⁻¹ (x + 1) + c

∫[(x²)/(9 + x⁶)] dx

∫[(x²)/(9 + x⁶)] dx = ∫[(x²)/(9 + (x³)²)] dx
u = x³
du = 3x² dx
∫[(x²)/(9 + x⁶)] dx = [1/3] ∫[1/(3² + (x³)²)] 3x² dx
∫[(x²)/(9 + x⁶)] dx = [1/3] ∫[1/(3² + u²)] dx
∫[(x²)/(9 + x⁶)] dx = [1/3] [1/3] tan⁻¹ [u/3] + c

∫[(x²)/(9 + x⁶)] dx = [1/9] tan⁻¹ [(x³)/3] + c

∫[1/(x² + 2x + 3)] dx

∫[1/(x² + 2x + 3)] dx = ∫[1/(x² + 2x + 1 + 2)] dx
∫[1/(x² + 2x + 3)] dx = ∫[1/((x + 1)² + √2²)]
u = x + 1
du = dx
∫[1/(x² + 2x + 3)] dx = ∫[1/(√2² + u²)] du
∫[1/(x² + 2x + 3)] dx = [1/(√2)] tan⁻¹ [u/(√2)] + c

∫[1/(x² + 2x + 3)] dx = [1/(√2)] tan⁻¹ [(x + 1)/(√2)] + c

∫[1/(√{4x² − (x lnx)²})] dx

∫[1/(√{4x² − (x lnx)²})] dx = ∫[1/(√{x²(2² − (lnx)²})] dx
∫[1/(√{4x² − (x lnx)²})] dx = ∫[1/x] [1/(√{2² − (lnx)²})] dx
u = lnx
du = [1/x] dx
∫[1/(√{4x² − (x lnx)²})] dx = ∫[1/(√{2² − u²})] du
∫[1/(√{4x² − (x lnx)²})] dx = sin⁻¹ [u/2] + c

∫[1/(√{4x² − (x lnx)²})] dx = sin⁻¹ [lnx/2] + c

∫[sinx/(√{16 − cos²(x)})] dx

u = cosx
du = −sinx dx
∫[sinx/(√{16 − cos²(x)})] dx = − ∫[1/(√{16 − cos²(x)})] (−sinx) dx
∫[sinx/(√{16 − cos²(x)})] dx = − ∫[1/(√{4² − u²})] du
∫[sinx/(√{16 − cos²(x)})] dx = − sin⁻¹ [u/2] + c

∫[sinx/(√{16 − cos²(x)})] dx = − sin⁻¹ [cosx/2] + c

∫[1/(x ln(5x)√{(ln(5x))² − 25})] dx

u = 5x
du = 5 dx
∫[1/(x ln(5x)√{(ln(5x))² − 25})] dx = [1/5] ∫[1/(x lnu√{(lnu)² − 5²})] du
∫[1/(x ln(5x)√{(ln(5x))² − 25})] dx = [1/5] ∫[5/(u lnu √{(lnu)² − 5²})] du
v = lnu
dv = [1/u] du
∫[1/(x ln(5x)√{(ln(5x))² − 25})] dx = ∫[1/(v √{v² − 5²})] dv
∫[1/(x ln(5x)√{(ln(5x))² − 25})] dx = [1/5] sec⁻¹ [v/5] + c

∫[1/(x ln(5x)√{(ln(5x))² − 25})] dx = [1/5] sec⁻¹ [ln5x/5] + c

∫[(x³)/((x⁴ + 1)² + 36)] dx

u = x⁴ + 1
du = 4x³ dx
∫[(x³)/((x⁴ + 1)² + 36)] dx = [1/4] ∫[1/((x⁴ + 1)² + 36)] 3x³ dx
∫[(x³)/((x⁴ + 1)² + 36)] dx = [1/4] ∫[1/((u² + 6²)] dx
∫[(x³)/((x⁴ + 1)² + 36)] dx = [1/4] [1/6] tan⁻¹ [u/6] + c

∫[(x³)/((x⁴ + 1)² + 36)] dx = [1/24] tan⁻¹ [(x⁴ + 1)/6] + c

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Answer

Integrals of Inverse Trigonometric Functions

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Intro

Types of Integrals: Inverse Trig Functions

One Property

Example 1

Example 2

Example 3

Example 4

Intro 0:00
Types of Integrals: Inverse Trig Functions 0:09

One Property

Example 1 1:19
Example 2 3:44
Example 3 4:53
Example 4 5:53

Mathematics: AP Calculus AB

Section 1: 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
Section 2: 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
Section 3: 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
Section 4: 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
Section 5: 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
Section 6: 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
Section 7: 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

Related Books

	Cracking the AP Calculus AB Exam, 2015 Edition Author: Princeton Review ISBN: 804124809 Publisher: Princeton Review Year: 2014 The book includes 3 full length practice tests with detailed explanations, a review of all the key concepts, and targeted strateeies to ace th exam for your highest score.
	Cliffs AP: Calculus AB & BC, 3rd Edition Authors: Dale W Johnson, Kerry J King ISBN: 764586831 Publisher: Cliffs Notes Year: 2001 The book has a topic-by topic breakdown and lots of problem approach suggestions for both free response and multiple choice calculus questions. The book also includes two full length tests.

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John Zhu

Integrals of Inverse Trigonometric Functions

Slide Duration:

Table of Contents

Section 1: Functions

Definitions & Properties of Functions

11m 26s

Intro