**Unformatted text preview: **SMU Classification: Restricted Part 2: Calculus
B: Integral Calculus
1.Integration Techniques
2.Applications
Version: 1.0
Date: Oct 20, 2018
Textbook, Chapter 5, Chapter 6 Grace in Mathematics Associate Professor Christopher Ting Page 1 of 44 SMU Classification: Restricted Grace in Mathematics Associate Professor Christopher Ting Page 2 of 44 SMU Classification: Restricted 3 ′
2 =
→ = =
3 = 2 is called the derivative of = Q: = 3
.
3 is called the ______?______ of = .
Grace in Mathematics Associate Professor Christopher Ting Page 3 of 44 SMU Classification: Restricted Q: = Q: = Textbook, p. 338 is called the ______?______ of = . is called a/an _____?_______ of = . + Grace in Mathematics ′ = , −π ′ = . Associate Professor Christopher Ting Page 4 of 44 SMU Classification: Restricted Antiderivatives and Indefinite Integrals Textbook, pp. 338-340 Grace in Mathematics Associate Professor Christopher Ting Page 5 of 44 SMU Classification: Restricted integrand integral sign Indicates that the
antidifferentiation
is performed with
respect to the
variable x.
Grace in Mathematics constant of
integration to represent the family of all
antiderivatives of f (x) indefinite
integral Associate Professor Christopher Ting Page 6 of 44 SMU Classification: Restricted න = () 1. The derivative of the antiderivative of () is (). ′ න = + 2. The antiderivative of the derivative of () is + .
Grace in Mathematics Associate Professor Christopher Ting Page 7 of 44 SMU Classification: Restricted Q: How to find all
the antiderivatives
2
of (), say () = ?
Grace in Mathematics Associate Professor Christopher Ting Page 8 of 44 SMU Classification: Restricted FORMULAS (Indefinite integrals of elementary functions)
න = 1 +1 + +1 න = + න =
+
ln 1
න = ln || + න sin = − cos + න cos = sin + Textbook, pp. 340, 525 1
n 1 1 =0 2 = −1 3 = 4 = ln 5 1
ln = 1
ln || = , ≠ 0 6 1
log = ln 1 1
log || =
, ≠ 0 ln 2 3
b 0,b 1 4 > 0, ≠ 1 5 6 Grace in Mathematics > 0, ≠ 1 7 sin = cos 8 cos = − sin These formulas will be
given in the “Formula
Table” in the final exam. Associate Professor Christopher Ting Page 9 of 44 Textbook, p. 341 SMU Classification: Restricted (constant multiple property) (sum and difference property) Grace in Mathematics Associate Professor Christopher Ting Page 10 of 44 SMU Classification: Restricted PROPERTIES (of indefinite integrals) ∙ =∙ න ∙ = ∙ න ()
න 2 = 2 ∙ න 1 = 2 + = 2 + 2 = 2 + Textbook, pp. 341-342 1= 0 න 0 1
= 0+1 + 0+1 Grace in Mathematics Associate Professor Christopher Ting Page 11 of 44 SMU Classification: Restricted PROPERTIES (of indefinite integrals, continued) ± () = ± න ± () = න () ± න () 1
1 න +
= න + න = + 1 + ln + 2 = + ln || + Textbook, pp. 341-342 Grace in Mathematics Associate Professor Christopher Ting Page 12 of 44 SMU Classification: Restricted Examples/Exercises In-Class Exercise Q122 න 5 CHECK! = 5 + න 9 CHECK! = 9 + CHECK! 5 8
= +
8 න(4 3 + 2 − 1) CHECK! = 4 + 2 − + 3 න 2 + CHECK! = 2 + 3 ln || + න 5 7 Textbook, p. 342, EXAMPLE 2 Grace in Mathematics Associate Professor Christopher Ting Page 13 of 44 SMU Classification: Restricted Examples/Exercises
4 න 3 CHECK! In-Class Exercise Q123
= −2 −2 + change power functions in denominator to numerator න5 ∙ 3 2 CHECK! = 35/3 + change radical forms to exponent forms
3 − 3 න 2 CHECK! 1
= 2 + 3 −1 + 2 rewrite quotients to sums, using fractions’ property න 2
− 6 3 2 න + 2 CHECK! = 3 2/3 − 4 3/2 + CHECK! 4
=
+ 2 + 4 rewrite products to sums, by removing parentheses
Associate Professor Christopher Ting
Grace in Mathematics
Textbook, p. 343, EXAMPLE 3 Page 14 of 44 SMU Classification: Restricted Examples/Exercises In-Class Exercise Q124 න 2 cos() CHECK! = 2 sin() + න 2 sin() + ln 2 cos CHECK! Grace in Mathematics = − 2 cos() + ln 2 sin + Associate Professor Christopher Ting Page 15 of 44 SMU Classification: Restricted Integration Technique I (integration by substitution)
Reversing the Chain Rule න ′ [ ] = ′ [ ] ∙ ′() ∙ ′ = + න ′ () = + = = substitution = ′ differential + Remarks: In general, we apply this technique when we see a composite function of basic
functions in the integrand. Hope ′() is a basic function of .
Textbook, pp. 349-353 Grace in Mathematics Associate Professor Christopher Ting Page 16 of 44 Textbook, p. 351 SMU Classification: Restricted Grace in Mathematics Associate Professor Christopher Ting Page 17 of 44 SMU Classification: Restricted Examples/Exercises න 3 + 4 10 3 +1 1. න ′ =
+ , ≠ −1
+1 න 2 2 1
2 න
3 3
1+
Textbook, p. 351, EXAMPLE 1 2. න ′ = + 1 ′
3. න = ln | | + () Grace in Mathematics Associate Professor Christopher Ting Page 18 of 44 Textbook, p. 351, EXAMPLE 1 SMU Classification: Restricted In-Class Exercise Q125 Grace in Mathematics Associate Professor Christopher Ting Page 19 of 44 Textbook, pp. 353-354, EXAMPLE 4 SMU Classification: Restricted In-Class Exercise Q126 ′ = → = Grace in Mathematics Associate Professor Christopher Ting Page 20 of 44 Textbook, p. 354, EXAMPLE 5 SMU Classification: Restricted In-Class Exercise Q127-Q129 Grace in Mathematics Associate Professor Christopher Ting Page 21 of 44 Textbook, p. 354, EXAMPLE 5(A) SMU Classification: Restricted In-Class Exercise Q127 = → = Grace in Mathematics Associate Professor Christopher Ting Page 22 of 44 SMU Classification: Restricted In-Class Exercise Q128 = − → = − Textbook, p. 355, EXAMPLE 5(B) Grace in Mathematics Associate Professor Christopher Ting Page 23 of 44 Textbook, p. 355, EXAMPLE 5(C) SMU Classification: Restricted In-Class Exercise Q129 = → = Grace in Mathematics Associate Professor Christopher Ting Page 24 of 44 SMU Classification: Restricted In-Class Exercise Q130 න 3 + 4 10 2 න 2 න 3
1+ Grace in Mathematics Associate Professor Christopher Ting Page 25 of 44 SMU Classification: Restricted An Alternative Approach In-Class Exercise Q131 න sin( 2 ) = 2 , = 2 ′ = 2
1
1
1
2
= න 2 sin( ) = න sin() = − cos() + 2
2
2
CHECK! Using the previous approach, we compute it as follows.
=
= න sin 2 , = 2 ′ 1
= − cos 2 + 2 = 2 → = 1
1
1 = න sin() = − cos() + 2
2
2
Grace in Mathematics Associate Professor Christopher Ting Page 26 of 44 SMU Classification: Restricted In-Class Exercise Q132 5 න[sin 2 ] cos(2) Grace in Mathematics Associate Professor Christopher Ting Page 27 of 44 SMU Classification: Restricted Additional Substitution Techniques
න In-Class Exercise Q133 +2 Method 1:
=න න =න +2−2
+2
+2
+2 = + 2 → = , = − 2 − න න + 2
= න + 2 − න න 1 +2 Method 2:
2
+2
2
+2 −2
=න = න 1/2 − 2−1/2 =න +2 1/2 2
= +2
3 3/2 Textbook, p. 356, EXAMPLE 6 − 2 න + 2
−4 +2 1/2 −1/2 + Grace in Mathematics 1
2 3/2
= − 42 + 3
2
= + 2 3/2 − 4 + 2
3 1/2 + Associate Professor Christopher Ting Page 28 of 44 SMU Classification: Restricted Integration Techniques II (integration by partial fractions)
1
න ( + 1)( + 2)
1
1
=න
− ( + 1) ( + 2)
1
1
=න − න +1
+2 = ln + 1 − ln + 2 + +1
= ln
+
+2 In-Class Exercise Q134
Partial Fractions
1
1
1
=
−
???
( + 1)( + 2) + 1 + 2
CHECK!
Step 1: Write down the template:
1 =
+
( + 1)( + 2) + 1 + 2
Step 2: Multiply ( + 1)( + 2) on both sides.
1 = ( + 2) + ( + 1) Step 3: Let = −2, and solve for .
1 = −2 + 2 + −2 + 1 → = −1
Step 4: Let = −1, and solve for .
1 = −1 + 2 + −1 + 1 → = 1 Grace in Mathematics Associate Professor Christopher Ting Page 29 of 44 SMU Classification: Restricted + =
+
(1 + 1 )(2 + 2 ) 1 + 1 2 + 2
Homework: In-Class Exercise Q135 2 + 3
න 2 =? − 3 − 40 +3
න 2 =? − 3 − 40 [Hint: 2 − 3 − 40 = − 8 + 5 ]
Grace in Mathematics Associate Professor Christopher Ting Page 30 of 44 SMU Classification: Restricted Integration Techniques III (integration by parts)
Reversing the Product Rule Textbook, p. 421 = ′ + ′ න ′ = − න ′ = = ′ = න = − න Grace in Mathematics = ′ Associate Professor Christopher Ting Page 31 of 44 SMU Classification: Restricted In-Class Exercise Q136 Q: Will both choices work?
Textbook, pp. 421-422, EXAMPLE 1 Grace in Mathematics Associate Professor Christopher Ting Page 32 of 44 SMU Classification: Restricted = → = ′ = 1 2 = → = න = + 2
Q: Is this simpler than the
original indefinite integral? 1 2 = න 2
Grace in Mathematics Associate Professor Christopher Ting Page 33 of 44 Textbook, p. 422 SMU Classification: Restricted = → = ′ = = → = න = + = න Q: Is it simple enough to compute?
Grace in Mathematics Associate Professor Christopher Ting Page 34 of 44 Textbook, p. 422 SMU Classification: Restricted Grace in Mathematics Associate Professor Christopher Ting Page 35 of 44 Textbook, p. 422, EXAMPLE 2 SMU Classification: Restricted In-Class Exercise Q137 =? =?
Grace in Mathematics Associate Professor Christopher Ting Page 36 of 44 Textbook, p. 423 SMU Classification: Restricted Any tips? Grace in Mathematics Associate Professor Christopher Ting Page 37 of 44 Textbook, p. 423 SMU Classification: Restricted : L
Grace in Mathematics ogarithmic >A lgebraic >E xponential Associate Professor Christopher Ting Page 38 of 44 SMU Classification: Restricted EXAMPLE 1 EXAMPLE 2 =______________________ න =_____________________ න ln =______________________ =_____________________ =______________________
x^2 EXAMPLE 3 න 2 − =_____________________
e^(-x) In-Class Exercise Q138 =______________________
ln(x) EXAMPLE 4
In-Class Exercise Q139
Textbook, pp. 421-425 න ln =_____________________
1 Grace in Mathematics Associate Professor Christopher Ting Page 39 of 44 SMU Classification: Restricted Example: A product involving trigonometric functions
In-Class Exercise Q140 න sin() න cos() Homework ′ = → = = = sin() → = න sin() = − cos() + = − cos() − න(− cos()) = − cos() + sin() + Choose for U : L >A ogarithmic lgebraic Grace in Mathematics >T >E rigonometric xponential Associate Professor Christopher Ting Page 40 of 44 SMU Classification: Restricted In-Class Exercise Q141 = sin() , = → = cos() , = Double integration by parts = sin() − න cos() = cos() , = → = −sin() , = න sin() = sin() − cos() − න − sin() න cos() = sin() − cos() − න sin Q: Is this a dead loop? Homework
Grace in Mathematics Associate Professor Christopher Ting Page 41 of 44 SMU Classification: Restricted (continued) න sin() = sin() − cos() − න sin() + න sin() + න sin() 2 න sin() = sin() − cos Grace in Mathematics Associate Professor Christopher Ting Page 42 of 44 SMU Classification: Restricted sin()
− cos() න sin() =
+
2 Grace in Mathematics Associate Professor Christopher Ting Page 43 of 44 Takeaways SMU Classification: Restricted Integration by substitution Grace in Mathematics Associate Professor Christopher Ting Page 44 of 44 ...

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