SlideShare a Scribd company logo
Integrating trigonometric functions
Ex. 7 p.335 u-Substitution and the Log Rule We can solve differential equations using the log rule as well. Solve the differential equation  Solution - separate y things from x things and integrate both sides.  Put the “plus C” on right side only. There are three basic choices for u:  u = x,  u = x ln x, and u = ln x.  The first two don’t fit the u’/u pattern.  If I rewrite the function to be  the pattern fits because u = ln x and du = (1/x)dx Rewrite with u-substitution: Back-substitute:
Up until now, we didn’t know how to integrate tan x, cot x, sec x, and csc x.  With the Log rule, we can now do integration of these functions. Ex 8 p. 336 Using a trig identity to integrate using log rule Find  Rewrite with trig identity: Let u = cos x.  Then du = – sin x dx
Ex 9 p. 336  Derivative of the secant function This problem needs a creative step, multiplying and dividing by the same quantity to make it work. Find Let u = sec x + tan x. (the denominator)  Then du = sec x tan x + sec 2  x, which is the numerator! Integrate and back-substitute
The other two are left as problems in the homework. I remember these log ones by realizing if they are co- things, they have a negative in front.  Secant and tangent go together, as do cosecant and cotangent. These can be written in different forms,  see #83-86
Ex 10 p. 337 Integrating Trigonometric Functions Evaluate  Using Pythagorean Identity, 1 + cot 2  x = csc 2  x Be careful with parentheses if graphing and getting definite integral with the calculator.
Last but not least, Ex 11 p. 337 Finding an average value Recall that average value refers to average height, which would be area divided by width.  Find the average value of f(x) = tan x on the interval [0,  π / 4]
1 π /4 y = 0.441 is avg value
5.2b p. 338 29-53 every other odd (skip 45), 65,70, 73, 83, 85

More Related Content

What's hot

Roots of polynomials
Roots of polynomialsRoots of polynomials
Roots of polynomialsdaferro
 
logarithms
logarithmslogarithms
logarithms
srivaralakshmi
 
1557 logarithm
1557 logarithm1557 logarithm
1557 logarithm
Dr Fereidoun Dejahang
 
Exponents Rules
Exponents RulesExponents Rules
Exponents Rules
BigPassy2
 
Parallel Lines 2
Parallel Lines 2Parallel Lines 2
Parallel Lines 2jennr21
 
Otter 2014-12-08-02
Otter 2014-12-08-02Otter 2014-12-08-02
Otter 2014-12-08-02
Ruo Ando
 
Properties of logarithms
Properties of logarithmsProperties of logarithms
Properties of logarithmsJessica Garcia
 
Math130 ch09
Math130 ch09Math130 ch09
Math130 ch09
Putrace
 
Exercise roots of equations
Exercise roots of equationsExercise roots of equations
Exercise roots of equationsDUBAN CASTRO
 
3.4 Polynomial Functions and Their Graphs
3.4 Polynomial Functions and Their Graphs3.4 Polynomial Functions and Their Graphs
3.4 Polynomial Functions and Their Graphs
smiller5
 
3.1 b solving systems graphically
3.1 b solving systems graphically3.1 b solving systems graphically
3.1 b solving systems graphicallyfthrower
 
Logarithms and exponents solve equations
Logarithms and exponents solve equationsLogarithms and exponents solve equations
Logarithms and exponents solve equationssrobbins4
 
4.3 Logarithmic Functions
4.3 Logarithmic Functions4.3 Logarithmic Functions
4.3 Logarithmic Functions
smiller5
 
4.4 Set operations on relations
4.4 Set operations on relations4.4 Set operations on relations
4.4 Set operations on relations
Jan Plaza
 
4.2 Multiplying Matrices
4.2 Multiplying Matrices4.2 Multiplying Matrices
4.2 Multiplying Matriceshisema01
 

What's hot (16)

Roots of polynomials
Roots of polynomialsRoots of polynomials
Roots of polynomials
 
logarithms
logarithmslogarithms
logarithms
 
1557 logarithm
1557 logarithm1557 logarithm
1557 logarithm
 
Exponents Rules
Exponents RulesExponents Rules
Exponents Rules
 
Parallel Lines 2
Parallel Lines 2Parallel Lines 2
Parallel Lines 2
 
Otter 2014-12-08-02
Otter 2014-12-08-02Otter 2014-12-08-02
Otter 2014-12-08-02
 
Properties of logarithms
Properties of logarithmsProperties of logarithms
Properties of logarithms
 
Math130 ch09
Math130 ch09Math130 ch09
Math130 ch09
 
Exercise roots of equations
Exercise roots of equationsExercise roots of equations
Exercise roots of equations
 
3.4 Polynomial Functions and Their Graphs
3.4 Polynomial Functions and Their Graphs3.4 Polynomial Functions and Their Graphs
3.4 Polynomial Functions and Their Graphs
 
3.1 b solving systems graphically
3.1 b solving systems graphically3.1 b solving systems graphically
3.1 b solving systems graphically
 
Roots of polynomials
Roots of polynomialsRoots of polynomials
Roots of polynomials
 
Logarithms and exponents solve equations
Logarithms and exponents solve equationsLogarithms and exponents solve equations
Logarithms and exponents solve equations
 
4.3 Logarithmic Functions
4.3 Logarithmic Functions4.3 Logarithmic Functions
4.3 Logarithmic Functions
 
4.4 Set operations on relations
4.4 Set operations on relations4.4 Set operations on relations
4.4 Set operations on relations
 
4.2 Multiplying Matrices
4.2 Multiplying Matrices4.2 Multiplying Matrices
4.2 Multiplying Matrices
 

Viewers also liked

Ap and dual enrollment presentation final
Ap and dual enrollment presentation   finalAp and dual enrollment presentation   final
Ap and dual enrollment presentation final
hartcher
 
Calc 8.7 l'hopital
Calc 8.7 l'hopitalCalc 8.7 l'hopital
Calc 8.7 l'hopitalhartcher
 
2nd tri calculus flashcards
2nd tri calculus flashcards2nd tri calculus flashcards
2nd tri calculus flashcardshartcher
 
7.4 A arc length
7.4 A arc length7.4 A arc length
7.4 A arc lengthhartcher
 
Lecture 17 optimization - section 4.6
Lecture 17   optimization - section 4.6Lecture 17   optimization - section 4.6
Lecture 17 optimization - section 4.6
njit-ronbrown
 
Ap and Dual Enrollment Presentation
Ap and Dual Enrollment PresentationAp and Dual Enrollment Presentation
Ap and Dual Enrollment Presentation
hartcher
 
Calc 8.7 again
Calc 8.7 againCalc 8.7 again
Calc 8.7 againhartcher
 

Viewers also liked (20)

Ap and dual enrollment presentation final
Ap and dual enrollment presentation   finalAp and dual enrollment presentation   final
Ap and dual enrollment presentation final
 
Calc 4.4b
Calc 4.4bCalc 4.4b
Calc 4.4b
 
Calc 4.5a
Calc 4.5aCalc 4.5a
Calc 4.5a
 
Calc 8.7 l'hopital
Calc 8.7 l'hopitalCalc 8.7 l'hopital
Calc 8.7 l'hopital
 
2nd tri calculus flashcards
2nd tri calculus flashcards2nd tri calculus flashcards
2nd tri calculus flashcards
 
Calc 7.3a
Calc 7.3aCalc 7.3a
Calc 7.3a
 
Calc 5.7
Calc 5.7Calc 5.7
Calc 5.7
 
Calc 2.3b
Calc 2.3bCalc 2.3b
Calc 2.3b
 
7.4 A arc length
7.4 A arc length7.4 A arc length
7.4 A arc length
 
Calc 7.2a
Calc 7.2aCalc 7.2a
Calc 7.2a
 
Lecture 17 optimization - section 4.6
Lecture 17   optimization - section 4.6Lecture 17   optimization - section 4.6
Lecture 17 optimization - section 4.6
 
Ap and Dual Enrollment Presentation
Ap and Dual Enrollment PresentationAp and Dual Enrollment Presentation
Ap and Dual Enrollment Presentation
 
Calc 5.4b
Calc 5.4bCalc 5.4b
Calc 5.4b
 
Calc 5.4a
Calc 5.4aCalc 5.4a
Calc 5.4a
 
Calc 4.5b
Calc 4.5bCalc 4.5b
Calc 4.5b
 
Calc 5.2b
Calc 5.2bCalc 5.2b
Calc 5.2b
 
Calc 5.1b
Calc 5.1bCalc 5.1b
Calc 5.1b
 
Calc 4.5a
Calc 4.5aCalc 4.5a
Calc 4.5a
 
Calc 5.2a
Calc 5.2aCalc 5.2a
Calc 5.2a
 
Calc 8.7 again
Calc 8.7 againCalc 8.7 again
Calc 8.7 again
 

Similar to Calc 5.2b

1543 integration in mathematics b
1543 integration in mathematics b1543 integration in mathematics b
1543 integration in mathematics b
Dr Fereidoun Dejahang
 
02 Notes Divide and Conquer
02 Notes Divide and Conquer02 Notes Divide and Conquer
02 Notes Divide and Conquer
Andres Mendez-Vazquez
 
13 1 basics_integration
13 1 basics_integration13 1 basics_integration
13 1 basics_integration
ManarAdham
 
Integral Calculus Anti Derivatives reviewer
Integral Calculus Anti Derivatives reviewerIntegral Calculus Anti Derivatives reviewer
Integral Calculus Anti Derivatives reviewer
JoshuaAgcopra
 
دالة الاكسبونيشل الرياضية والفريدة من نوعها و كيفية استخدامهال
دالة الاكسبونيشل الرياضية والفريدة من نوعها و كيفية استخدامهالدالة الاكسبونيشل الرياضية والفريدة من نوعها و كيفية استخدامهال
دالة الاكسبونيشل الرياضية والفريدة من نوعها و كيفية استخدامهال
zeeko4
 
5.3 Graphs of Polynomial Functions
5.3 Graphs of Polynomial Functions5.3 Graphs of Polynomial Functions
5.3 Graphs of Polynomial Functions
smiller5
 
Developing Expert Voices
Developing Expert VoicesDeveloping Expert Voices
Developing Expert Voicessuzanne
 
integration in maths pdf mathematics integration
integration in maths pdf mathematics integrationintegration in maths pdf mathematics integration
integration in maths pdf mathematics integration
Dr. Karrar Alwash
 
Maths Revision - GCSE And Additional Notes
Maths Revision - GCSE And Additional NotesMaths Revision - GCSE And Additional Notes
Maths Revision - GCSE And Additional NotesKatie B
 
The Fundamental theorem of calculus
The Fundamental theorem of calculus The Fundamental theorem of calculus
The Fundamental theorem of calculus
AhsanIrshad8
 
P1-Chp13-Integration.pptx
P1-Chp13-Integration.pptxP1-Chp13-Integration.pptx
P1-Chp13-Integration.pptx
YashTiwari512330
 
20230411 Discussion of Integration by Substitution.docx
20230411 Discussion of Integration by Substitution.docx20230411 Discussion of Integration by Substitution.docx
20230411 Discussion of Integration by Substitution.docx
Sharon Liu
 
Calculus 08 techniques_of_integration
Calculus 08 techniques_of_integrationCalculus 08 techniques_of_integration
Calculus 08 techniques_of_integrationtutulk
 
Maths AIP.pdf
Maths AIP.pdfMaths AIP.pdf
Maths AIP.pdf
YuvrajSingh835126
 
Calculus - Functions Review
Calculus - Functions ReviewCalculus - Functions Review
Calculus - Functions Reviewhassaanciit
 
Natural Logs
Natural LogsNatural Logs
Natural Logsswartzje
 
Section 7.5
Section 7.5Section 7.5
Section 7.5
CalculusII
 
Exp integrals
Exp integralsExp integrals
INTEGRATION-1.pptx
INTEGRATION-1.pptxINTEGRATION-1.pptx
INTEGRATION-1.pptx
SayanSen36
 

Similar to Calc 5.2b (20)

1543 integration in mathematics b
1543 integration in mathematics b1543 integration in mathematics b
1543 integration in mathematics b
 
02 Notes Divide and Conquer
02 Notes Divide and Conquer02 Notes Divide and Conquer
02 Notes Divide and Conquer
 
13 1 basics_integration
13 1 basics_integration13 1 basics_integration
13 1 basics_integration
 
Integral Calculus Anti Derivatives reviewer
Integral Calculus Anti Derivatives reviewerIntegral Calculus Anti Derivatives reviewer
Integral Calculus Anti Derivatives reviewer
 
دالة الاكسبونيشل الرياضية والفريدة من نوعها و كيفية استخدامهال
دالة الاكسبونيشل الرياضية والفريدة من نوعها و كيفية استخدامهالدالة الاكسبونيشل الرياضية والفريدة من نوعها و كيفية استخدامهال
دالة الاكسبونيشل الرياضية والفريدة من نوعها و كيفية استخدامهال
 
5.3 Graphs of Polynomial Functions
5.3 Graphs of Polynomial Functions5.3 Graphs of Polynomial Functions
5.3 Graphs of Polynomial Functions
 
Developing Expert Voices
Developing Expert VoicesDeveloping Expert Voices
Developing Expert Voices
 
integration in maths pdf mathematics integration
integration in maths pdf mathematics integrationintegration in maths pdf mathematics integration
integration in maths pdf mathematics integration
 
Maths Revision - GCSE And Additional Notes
Maths Revision - GCSE And Additional NotesMaths Revision - GCSE And Additional Notes
Maths Revision - GCSE And Additional Notes
 
The Fundamental theorem of calculus
The Fundamental theorem of calculus The Fundamental theorem of calculus
The Fundamental theorem of calculus
 
P1-Chp13-Integration.pptx
P1-Chp13-Integration.pptxP1-Chp13-Integration.pptx
P1-Chp13-Integration.pptx
 
20230411 Discussion of Integration by Substitution.docx
20230411 Discussion of Integration by Substitution.docx20230411 Discussion of Integration by Substitution.docx
20230411 Discussion of Integration by Substitution.docx
 
Calculus 08 techniques_of_integration
Calculus 08 techniques_of_integrationCalculus 08 techniques_of_integration
Calculus 08 techniques_of_integration
 
The integral
The integralThe integral
The integral
 
Maths AIP.pdf
Maths AIP.pdfMaths AIP.pdf
Maths AIP.pdf
 
Calculus - Functions Review
Calculus - Functions ReviewCalculus - Functions Review
Calculus - Functions Review
 
Natural Logs
Natural LogsNatural Logs
Natural Logs
 
Section 7.5
Section 7.5Section 7.5
Section 7.5
 
Exp integrals
Exp integralsExp integrals
Exp integrals
 
INTEGRATION-1.pptx
INTEGRATION-1.pptxINTEGRATION-1.pptx
INTEGRATION-1.pptx
 

More from hartcher

Binomial distributions
Binomial distributionsBinomial distributions
Binomial distributions
hartcher
 
10.2 using combinations and the binomial theorem
10.2 using combinations and the binomial theorem10.2 using combinations and the binomial theorem
10.2 using combinations and the binomial theorem
hartcher
 
Calc 3.4b
Calc 3.4bCalc 3.4b
Calc 3.4b
hartcher
 
2.6b scatter plots and lines of best fit
2.6b scatter plots and lines of best fit2.6b scatter plots and lines of best fit
2.6b scatter plots and lines of best fit
hartcher
 
Ap and dual enrollment presentation
Ap and dual enrollment presentationAp and dual enrollment presentation
Ap and dual enrollment presentation
hartcher
 
AP and Dual Enrollment Presentation
AP and Dual Enrollment PresentationAP and Dual Enrollment Presentation
AP and Dual Enrollment Presentation
hartcher
 

More from hartcher (20)

Binomial distributions
Binomial distributionsBinomial distributions
Binomial distributions
 
10.2 using combinations and the binomial theorem
10.2 using combinations and the binomial theorem10.2 using combinations and the binomial theorem
10.2 using combinations and the binomial theorem
 
Calc 3.4b
Calc 3.4bCalc 3.4b
Calc 3.4b
 
2.6b scatter plots and lines of best fit
2.6b scatter plots and lines of best fit2.6b scatter plots and lines of best fit
2.6b scatter plots and lines of best fit
 
Ap and dual enrollment presentation
Ap and dual enrollment presentationAp and dual enrollment presentation
Ap and dual enrollment presentation
 
AP and Dual Enrollment Presentation
AP and Dual Enrollment PresentationAP and Dual Enrollment Presentation
AP and Dual Enrollment Presentation
 
Calc 2.2b
Calc 2.2bCalc 2.2b
Calc 2.2b
 
Calc 2.6
Calc 2.6Calc 2.6
Calc 2.6
 
Calc 6.1b
Calc 6.1bCalc 6.1b
Calc 6.1b
 
Calc 6.1a
Calc 6.1aCalc 6.1a
Calc 6.1a
 
Calc 7.3b
Calc 7.3bCalc 7.3b
Calc 7.3b
 
Calc 7.2b
Calc 7.2bCalc 7.2b
Calc 7.2b
 
Calc 7.1b
Calc 7.1bCalc 7.1b
Calc 7.1b
 
Calc 7.1a
Calc 7.1aCalc 7.1a
Calc 7.1a
 
Calc 5.8b
Calc 5.8bCalc 5.8b
Calc 5.8b
 
Calc 5.8a
Calc 5.8aCalc 5.8a
Calc 5.8a
 
Calc 5.6
Calc 5.6Calc 5.6
Calc 5.6
 
Calc 5.5b
Calc 5.5bCalc 5.5b
Calc 5.5b
 
Calc 5.5a
Calc 5.5aCalc 5.5a
Calc 5.5a
 
Calc 5.3
Calc 5.3Calc 5.3
Calc 5.3
 

Recently uploaded

Encryption in Microsoft 365 - ExpertsLive Netherlands 2024
Encryption in Microsoft 365 - ExpertsLive Netherlands 2024Encryption in Microsoft 365 - ExpertsLive Netherlands 2024
Encryption in Microsoft 365 - ExpertsLive Netherlands 2024
Albert Hoitingh
 
Pushing the limits of ePRTC: 100ns holdover for 100 days
Pushing the limits of ePRTC: 100ns holdover for 100 daysPushing the limits of ePRTC: 100ns holdover for 100 days
Pushing the limits of ePRTC: 100ns holdover for 100 days
Adtran
 
20240607 QFM018 Elixir Reading List May 2024
20240607 QFM018 Elixir Reading List May 202420240607 QFM018 Elixir Reading List May 2024
20240607 QFM018 Elixir Reading List May 2024
Matthew Sinclair
 
Introduction to CHERI technology - Cybersecurity
Introduction to CHERI technology - CybersecurityIntroduction to CHERI technology - Cybersecurity
Introduction to CHERI technology - Cybersecurity
mikeeftimakis1
 
Epistemic Interaction - tuning interfaces to provide information for AI support
Epistemic Interaction - tuning interfaces to provide information for AI supportEpistemic Interaction - tuning interfaces to provide information for AI support
Epistemic Interaction - tuning interfaces to provide information for AI support
Alan Dix
 
GraphSummit Singapore | Enhancing Changi Airport Group's Passenger Experience...
GraphSummit Singapore | Enhancing Changi Airport Group's Passenger Experience...GraphSummit Singapore | Enhancing Changi Airport Group's Passenger Experience...
GraphSummit Singapore | Enhancing Changi Airport Group's Passenger Experience...
Neo4j
 
By Design, not by Accident - Agile Venture Bolzano 2024
By Design, not by Accident - Agile Venture Bolzano 2024By Design, not by Accident - Agile Venture Bolzano 2024
By Design, not by Accident - Agile Venture Bolzano 2024
Pierluigi Pugliese
 
Observability Concepts EVERY Developer Should Know -- DeveloperWeek Europe.pdf
Observability Concepts EVERY Developer Should Know -- DeveloperWeek Europe.pdfObservability Concepts EVERY Developer Should Know -- DeveloperWeek Europe.pdf
Observability Concepts EVERY Developer Should Know -- DeveloperWeek Europe.pdf
Paige Cruz
 
Monitoring Java Application Security with JDK Tools and JFR Events
Monitoring Java Application Security with JDK Tools and JFR EventsMonitoring Java Application Security with JDK Tools and JFR Events
Monitoring Java Application Security with JDK Tools and JFR Events
Ana-Maria Mihalceanu
 
GridMate - End to end testing is a critical piece to ensure quality and avoid...
GridMate - End to end testing is a critical piece to ensure quality and avoid...GridMate - End to end testing is a critical piece to ensure quality and avoid...
GridMate - End to end testing is a critical piece to ensure quality and avoid...
ThomasParaiso2
 
Smart TV Buyer Insights Survey 2024 by 91mobiles.pdf
Smart TV Buyer Insights Survey 2024 by 91mobiles.pdfSmart TV Buyer Insights Survey 2024 by 91mobiles.pdf
Smart TV Buyer Insights Survey 2024 by 91mobiles.pdf
91mobiles
 
FIDO Alliance Osaka Seminar: Passkeys at Amazon.pdf
FIDO Alliance Osaka Seminar: Passkeys at Amazon.pdfFIDO Alliance Osaka Seminar: Passkeys at Amazon.pdf
FIDO Alliance Osaka Seminar: Passkeys at Amazon.pdf
FIDO Alliance
 
Secstrike : Reverse Engineering & Pwnable tools for CTF.pptx
Secstrike : Reverse Engineering & Pwnable tools for CTF.pptxSecstrike : Reverse Engineering & Pwnable tools for CTF.pptx
Secstrike : Reverse Engineering & Pwnable tools for CTF.pptx
nkrafacyberclub
 
PHP Frameworks: I want to break free (IPC Berlin 2024)
PHP Frameworks: I want to break free (IPC Berlin 2024)PHP Frameworks: I want to break free (IPC Berlin 2024)
PHP Frameworks: I want to break free (IPC Berlin 2024)
Ralf Eggert
 
FIDO Alliance Osaka Seminar: Passkeys and the Road Ahead.pdf
FIDO Alliance Osaka Seminar: Passkeys and the Road Ahead.pdfFIDO Alliance Osaka Seminar: Passkeys and the Road Ahead.pdf
FIDO Alliance Osaka Seminar: Passkeys and the Road Ahead.pdf
FIDO Alliance
 
Elevating Tactical DDD Patterns Through Object Calisthenics
Elevating Tactical DDD Patterns Through Object CalisthenicsElevating Tactical DDD Patterns Through Object Calisthenics
Elevating Tactical DDD Patterns Through Object Calisthenics
Dorra BARTAGUIZ
 
GDG Cloud Southlake #33: Boule & Rebala: Effective AppSec in SDLC using Deplo...
GDG Cloud Southlake #33: Boule & Rebala: Effective AppSec in SDLC using Deplo...GDG Cloud Southlake #33: Boule & Rebala: Effective AppSec in SDLC using Deplo...
GDG Cloud Southlake #33: Boule & Rebala: Effective AppSec in SDLC using Deplo...
James Anderson
 
Alt. GDG Cloud Southlake #33: Boule & Rebala: Effective AppSec in SDLC using ...
Alt. GDG Cloud Southlake #33: Boule & Rebala: Effective AppSec in SDLC using ...Alt. GDG Cloud Southlake #33: Boule & Rebala: Effective AppSec in SDLC using ...
Alt. GDG Cloud Southlake #33: Boule & Rebala: Effective AppSec in SDLC using ...
James Anderson
 
GraphSummit Singapore | The Art of the Possible with Graph - Q2 2024
GraphSummit Singapore | The Art of the  Possible with Graph - Q2 2024GraphSummit Singapore | The Art of the  Possible with Graph - Q2 2024
GraphSummit Singapore | The Art of the Possible with Graph - Q2 2024
Neo4j
 
Uni Systems Copilot event_05062024_C.Vlachos.pdf
Uni Systems Copilot event_05062024_C.Vlachos.pdfUni Systems Copilot event_05062024_C.Vlachos.pdf
Uni Systems Copilot event_05062024_C.Vlachos.pdf
Uni Systems S.M.S.A.
 

Recently uploaded (20)

Encryption in Microsoft 365 - ExpertsLive Netherlands 2024
Encryption in Microsoft 365 - ExpertsLive Netherlands 2024Encryption in Microsoft 365 - ExpertsLive Netherlands 2024
Encryption in Microsoft 365 - ExpertsLive Netherlands 2024
 
Pushing the limits of ePRTC: 100ns holdover for 100 days
Pushing the limits of ePRTC: 100ns holdover for 100 daysPushing the limits of ePRTC: 100ns holdover for 100 days
Pushing the limits of ePRTC: 100ns holdover for 100 days
 
20240607 QFM018 Elixir Reading List May 2024
20240607 QFM018 Elixir Reading List May 202420240607 QFM018 Elixir Reading List May 2024
20240607 QFM018 Elixir Reading List May 2024
 
Introduction to CHERI technology - Cybersecurity
Introduction to CHERI technology - CybersecurityIntroduction to CHERI technology - Cybersecurity
Introduction to CHERI technology - Cybersecurity
 
Epistemic Interaction - tuning interfaces to provide information for AI support
Epistemic Interaction - tuning interfaces to provide information for AI supportEpistemic Interaction - tuning interfaces to provide information for AI support
Epistemic Interaction - tuning interfaces to provide information for AI support
 
GraphSummit Singapore | Enhancing Changi Airport Group's Passenger Experience...
GraphSummit Singapore | Enhancing Changi Airport Group's Passenger Experience...GraphSummit Singapore | Enhancing Changi Airport Group's Passenger Experience...
GraphSummit Singapore | Enhancing Changi Airport Group's Passenger Experience...
 
By Design, not by Accident - Agile Venture Bolzano 2024
By Design, not by Accident - Agile Venture Bolzano 2024By Design, not by Accident - Agile Venture Bolzano 2024
By Design, not by Accident - Agile Venture Bolzano 2024
 
Observability Concepts EVERY Developer Should Know -- DeveloperWeek Europe.pdf
Observability Concepts EVERY Developer Should Know -- DeveloperWeek Europe.pdfObservability Concepts EVERY Developer Should Know -- DeveloperWeek Europe.pdf
Observability Concepts EVERY Developer Should Know -- DeveloperWeek Europe.pdf
 
Monitoring Java Application Security with JDK Tools and JFR Events
Monitoring Java Application Security with JDK Tools and JFR EventsMonitoring Java Application Security with JDK Tools and JFR Events
Monitoring Java Application Security with JDK Tools and JFR Events
 
GridMate - End to end testing is a critical piece to ensure quality and avoid...
GridMate - End to end testing is a critical piece to ensure quality and avoid...GridMate - End to end testing is a critical piece to ensure quality and avoid...
GridMate - End to end testing is a critical piece to ensure quality and avoid...
 
Smart TV Buyer Insights Survey 2024 by 91mobiles.pdf
Smart TV Buyer Insights Survey 2024 by 91mobiles.pdfSmart TV Buyer Insights Survey 2024 by 91mobiles.pdf
Smart TV Buyer Insights Survey 2024 by 91mobiles.pdf
 
FIDO Alliance Osaka Seminar: Passkeys at Amazon.pdf
FIDO Alliance Osaka Seminar: Passkeys at Amazon.pdfFIDO Alliance Osaka Seminar: Passkeys at Amazon.pdf
FIDO Alliance Osaka Seminar: Passkeys at Amazon.pdf
 
Secstrike : Reverse Engineering & Pwnable tools for CTF.pptx
Secstrike : Reverse Engineering & Pwnable tools for CTF.pptxSecstrike : Reverse Engineering & Pwnable tools for CTF.pptx
Secstrike : Reverse Engineering & Pwnable tools for CTF.pptx
 
PHP Frameworks: I want to break free (IPC Berlin 2024)
PHP Frameworks: I want to break free (IPC Berlin 2024)PHP Frameworks: I want to break free (IPC Berlin 2024)
PHP Frameworks: I want to break free (IPC Berlin 2024)
 
FIDO Alliance Osaka Seminar: Passkeys and the Road Ahead.pdf
FIDO Alliance Osaka Seminar: Passkeys and the Road Ahead.pdfFIDO Alliance Osaka Seminar: Passkeys and the Road Ahead.pdf
FIDO Alliance Osaka Seminar: Passkeys and the Road Ahead.pdf
 
Elevating Tactical DDD Patterns Through Object Calisthenics
Elevating Tactical DDD Patterns Through Object CalisthenicsElevating Tactical DDD Patterns Through Object Calisthenics
Elevating Tactical DDD Patterns Through Object Calisthenics
 
GDG Cloud Southlake #33: Boule & Rebala: Effective AppSec in SDLC using Deplo...
GDG Cloud Southlake #33: Boule & Rebala: Effective AppSec in SDLC using Deplo...GDG Cloud Southlake #33: Boule & Rebala: Effective AppSec in SDLC using Deplo...
GDG Cloud Southlake #33: Boule & Rebala: Effective AppSec in SDLC using Deplo...
 
Alt. GDG Cloud Southlake #33: Boule & Rebala: Effective AppSec in SDLC using ...
Alt. GDG Cloud Southlake #33: Boule & Rebala: Effective AppSec in SDLC using ...Alt. GDG Cloud Southlake #33: Boule & Rebala: Effective AppSec in SDLC using ...
Alt. GDG Cloud Southlake #33: Boule & Rebala: Effective AppSec in SDLC using ...
 
GraphSummit Singapore | The Art of the Possible with Graph - Q2 2024
GraphSummit Singapore | The Art of the  Possible with Graph - Q2 2024GraphSummit Singapore | The Art of the  Possible with Graph - Q2 2024
GraphSummit Singapore | The Art of the Possible with Graph - Q2 2024
 
Uni Systems Copilot event_05062024_C.Vlachos.pdf
Uni Systems Copilot event_05062024_C.Vlachos.pdfUni Systems Copilot event_05062024_C.Vlachos.pdf
Uni Systems Copilot event_05062024_C.Vlachos.pdf
 

Calc 5.2b

  • 2. Ex. 7 p.335 u-Substitution and the Log Rule We can solve differential equations using the log rule as well. Solve the differential equation Solution - separate y things from x things and integrate both sides. Put the “plus C” on right side only. There are three basic choices for u: u = x, u = x ln x, and u = ln x. The first two don’t fit the u’/u pattern. If I rewrite the function to be the pattern fits because u = ln x and du = (1/x)dx Rewrite with u-substitution: Back-substitute:
  • 3. Up until now, we didn’t know how to integrate tan x, cot x, sec x, and csc x. With the Log rule, we can now do integration of these functions. Ex 8 p. 336 Using a trig identity to integrate using log rule Find Rewrite with trig identity: Let u = cos x. Then du = – sin x dx
  • 4. Ex 9 p. 336 Derivative of the secant function This problem needs a creative step, multiplying and dividing by the same quantity to make it work. Find Let u = sec x + tan x. (the denominator) Then du = sec x tan x + sec 2 x, which is the numerator! Integrate and back-substitute
  • 5. The other two are left as problems in the homework. I remember these log ones by realizing if they are co- things, they have a negative in front. Secant and tangent go together, as do cosecant and cotangent. These can be written in different forms, see #83-86
  • 6. Ex 10 p. 337 Integrating Trigonometric Functions Evaluate Using Pythagorean Identity, 1 + cot 2 x = csc 2 x Be careful with parentheses if graphing and getting definite integral with the calculator.
  • 7. Last but not least, Ex 11 p. 337 Finding an average value Recall that average value refers to average height, which would be area divided by width. Find the average value of f(x) = tan x on the interval [0, π / 4]
  • 8. 1 π /4 y = 0.441 is avg value
  • 9. 5.2b p. 338 29-53 every other odd (skip 45), 65,70, 73, 83, 85