Inverse Trig Functions & Integration
© G. Battaly 2015 1
September 28, 2015
Calculus Home PageClass Notes: Prof. G. Battaly, Westchester Community College, NY
Homework Part 1
5.7 Inverse Trig Functions and Integration
HW: 5.7 # 5 13, 21 31, 47
GOAL:1. Distinguish methods of integrating rational expressions using past procedures.2. Recognize rational expressions that integrate to inverse trig functions.3. Perform integrations that involve inverse trig functions.
5.7 Inverse Trig Functions and Integration
Calculus Home PageClass Notes: Prof. G. Battaly, Westchester Community College, NY
Homework Part 1
Which can be integrated? How?
1. 2.
3. 4.
Inverse Trig Functions & Integration
© G. Battaly 2015 2
September 28, 2015
5.7 Inverse Trig Functions and Integration
Calculus Home PageClass Notes: Prof. G. Battaly, Westchester Community College, NY
Homework Part 1
Which can be integrated? How?
1. 2.
3. 4.
5.7 Inverse Trig Functions and Integration
Calculus Home PageClass Notes: Prof. G. Battaly, Westchester Community College, NY
Homework Part 1
Integrating Rational Expressions:1. Is the denominator a monomial?
> If yes, divide and integrate. > If no, proceed to #2.
2. If the quotient is rewritten as a product, is the exponent (1)? > If yes, think ln and look for du/u > If no, think un and look for un du
3. If none of above work, think inverse trig functions and look for du/ √a2 u2 or du/ ( a2 + u2 ) or du/ u √u2 a2
Previous
New
Inverse Trig Functions & Integration
© G. Battaly 2015 3
September 28, 2015
5.7 Inverse Trig Functions and Integration
Can we integrate this using ∫ un du ?
Calculus Home PageClass Notes: Prof. G. Battaly, Westchester Community College, NY
Homework Part 1
5.7 Inverse Trig Functions and Integration
Can we integrate this using ∫ un du ?
Does not work. The result is not less complicated than original. So there is no progress.
Calculus Home PageClass Notes: Prof. G. Battaly, Westchester Community College, NY
Homework Part 1
Inverse Trig Functions & Integration
© G. Battaly 2015 4
September 28, 2015
5.7 Inverse Trig Functions and Integration
Calculus Home PageClass Notes: Prof. G. Battaly, Westchester Community College, NY
Homework Part 1
Integrating Rational Expressions:1. Is the denominator a monomial?
> If yes, divide and integrate. > If no, proceed to #2.
2. If the quotient is rewritten as a product, is the exponent (1)? > If yes, think ln and look for du/u > If no, think un and look for un du
3. If none of above work, think inverse trig functions and look for du/ √a2 u2 or du/ ( a2 + u2 ) or du/ u √u2 a2
NO
NO
Not undu > no du
YES
5.7 Inverse Trig Functions and Integration
Calculus Home PageClass Notes: Prof. G. Battaly, Westchester Community College, NY
Homework Part 1
using trig:
Inverse Trig Functions & Integration
© G. Battaly 2015 5
September 28, 2015
5.7 Inverse Trig Functions and Integration
Is there a shorter way?
Calculus Home PageClass Notes: Prof. G. Battaly, Westchester Community College, NY
Homework Part 1
5.7 Inverse Trig Functions and Integration
What does this form suggest?
Calculus Home PageClass Notes: Prof. G. Battaly, Westchester Community College, NY
Homework Part 1
θ
Inverse Trig Functions & Integration
© G. Battaly 2015 6
September 28, 2015
5.7 Inverse Trig Functions and Integration
What does this form suggest?
Calculus Home PageClass Notes: Prof. G. Battaly, Westchester Community College, NY
Homework Part 1
hyp sidesinθ = 2x 1
θ
∴Use arc sin (___)
Calculus Home PageClass Notes: Prof. G. Battaly, Westchester Community College, NY
Homework Part 1
using formula:
5.7 Inverse Trig Functions and Integration
Inverse Trig Functions & Integration
© G. Battaly 2015 7
September 28, 2015
Calculus Home PageClass Notes: Prof. G. Battaly, Westchester Community College, NY
Homework Part 1
using formula:
5.7 Inverse Trig Functions and Integration
5.7 Inverse Trig Functions and Integration
Calculus Home PageClass Notes: Prof. G. Battaly, Westchester Community College, NY
Homework Part 1
Integrating Rational Expressions:1. Is the denominator a monomial?
> If yes, divide and integrate. > If no, proceed to #2.
2. If the quotient is rewritten as a product, is the exponent (1)? > If yes, think ln and look for du/u > If no, think un and look for un du
3. If none of above work, think inverse trig functions and look for du/ √a2 u2 or du/ ( a2 + u2 ) or du/ u √u2 a2
Inverse Trig Functions & Integration
© G. Battaly 2015 8
September 28, 2015
5.7 Inverse Trig Functions and Integration
Calculus Home PageClass Notes: Prof. G. Battaly, Westchester Community College, NY
Homework Part 1
Integrating Rational Expressions:1. Is the denominator a monomial?
> If yes, divide and integrate. > If no, proceed to #2.
2. If the quotient is rewritten as a product, is the exponent (1)? > If yes, think ln and look for du/u > If no, think un and look for un du
3. If none of above work, think inverse trig functions and look for du/ √a2 u2 or du/ ( a2 + u2 ) or du/ u √u2 a2
NO
Not du/u > no du
YES
yes
θtanθ = a b
∴Use arc tan (___)
5.7 Inverse Trig Functions and Integration
Calculus Home PageClass Notes: Prof. G. Battaly, Westchester Community College, NY
Homework Part 1
Inverse Trig Functions & Integration
© G. Battaly 2015 9
September 28, 2015
5.7 Inverse Trig Functions and Integration
Calculus Home PageClass Notes: Prof. G. Battaly, Westchester Community College, NY
Homework Part 1
Calculus Home PageClass Notes: Prof. G. Battaly, Westchester Community College, NY
Homework Part 1
5.7 Inverse Trig Functions and Integration
Inverse Trig Functions & Integration
© G. Battaly 2015 10
September 28, 2015
Calculus Home PageClass Notes: Prof. G. Battaly, Westchester Community College, NY
Homework Part 1
5.7 Inverse Trig Functions and Integration
Calculus Home PageClass Notes: Prof. G. Battaly, Westchester Community College, NY
Homework Part 1
5.7 Inverse Trig Functions and Integration
Inverse Trig Functions & Integration
© G. Battaly 2015 11
September 28, 2015
Calculus Home PageClass Notes: Prof. G. Battaly, Westchester Community College, NY
Homework Part 1
5.7 Inverse Trig Functions and Integration
Calculus Home PageClass Notes: Prof. G. Battaly, Westchester Community College, NY
Homework Part 1
5.7 Inverse Trig Functions and Integration formulas
Inverse Trig Functions & Integration
© G. Battaly 2015 12
September 28, 2015
Calculus Home PageClass Notes: Prof. G. Battaly, Westchester Community College, NY
Homework Part 1
5.7 Inverse Trig Functions and Integration formulas
u=arc cosxdu= dx √1x2 x u1/√2 π/4 0 π/2
5.7 Inverse Trig Functions and Integration
Calculus Home PageClass Notes: Prof. G. Battaly, Westchester Community College, NY
Homework Part 1
formulas
Inverse Trig Functions & Integration
© G. Battaly 2015 13
September 28, 2015