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Review Of Formulas And Techniques
Integration Table
EX 1.1 A Simple Substitution
EX 1.2 Generalizing a Basic Integration Rule
EX 1.3 An Integrand That Must Be Expanded
EX 1.4 An Integral Where We Must Complete the Square
EX 1.5 An Integral Requiring Some Imagination
Integration By Parts
Let
？
EX 2.1 Integration by Parts
Let
So
Remark
How about letting and
EX 2.3 An Integrand with a Single Term
Let
EX 2.4 Repeated Integration by Parts
EX 2.5 Repeated Integration by Parts with a Twist
Reduction Formula
EX 2.6 Using a Reduction Formula
…
Integration by Parts with A Definite Integral
EX 2.7 Integration by Parts for a Definite Integral
Integrals Involving Powers of Trigonometric Functions
Case 1: m or n Is an Odd Positive Integer
Case 2: m and n Are Both Even Positive Integers
Type A
EX 3.2 An Integrand with an Odd Power of Sine
Let
EX 3.3 An Integrand with an Odd Power of Cosine
Let
EX 3.4 An Integrand with an Even Power of Sine
EX 3.5 An Integrand with an Even Power of Cosine
Type B
Integrals Involving Powers of Trigonometric Functions
Case 1: m Is an Odd Positive Integer
Case 2: n Is an Even Positive Integer
Case 3: m Is an Even Positive Integer and n Is an Odd Positive Integer
EX 3.6 An Integrand with an Odd Power of Tangent
EX 3.7 An Integrand with an Even Power of Secant
EX 3.8 An Unusual Integral Evaluate
Trigonometric Substitution
How to integrate the following forms ?
for some a > 0
Case A
Case B
Case C
EX 3.9 An Integral Involving
EX 3.10 An Integral Involving
EX 3.11 An Integral Involving
Using Partial Fractions
Distinct linear factors
Repeated linear factors
Irreducible quadratic factors
EX 4.1 Partial Fractions: Distinct Linear Factors
EX 4.2 Partial Fractions: Three Distinct Linear Factors
EX 4.3 Partial Fractions Where Long Division Is Required
dxxf )(
EX 4.4 Partial Fractions with a Repeated Linear Factor
dxxf )(
EX 4.5 Partial Fractions with a Quadratic Factor
dxxf )(
EX 4.6 Partial Fractions with a Quadratic Factor
A=2 B=3 and C=4
dxxf )(
EX 4.6 (Cont’d)
Reasons
The Fundamental Theorem assumes a continuous integrand, our use of the theorem is invalid and our answer is incorrect.
1
2
EX 6.1 An Integrand That Blows Up at the Right Endpoint
EX 6.2 A Divergent Improper Integral
EX 6.3 A Convergent Improper Integral
EX 6.4 A Divergent Improper Integral
EX 6.5 An Integrand That Blows Up in the Middle of an Interval
EX 6.6 An Integral with an Infinite Limit of Integration
EX 6.7 A Divergent Improper Integral
EX 6.8 A Convergent Improper Integral
EX 6.9 An Integral with an Infinite Lower Limit of Integration
EX 6.10 A Convergent Improper Integral
Remark
It’s certainly tempting to write this, especially since this will often give a correct answer, with about half of the work.Unfortunately, this will often give incorrect answers, too, as the limit on the right-hand side frequently exists for divergent integrals.
EX 6.11 An Integral with Two Infinite Limits of Integration
EX 6.12 An Integral with Two Infinite Limits of Integration
EX 6.13 An Integral That Is Improper for Two Reasons
f(x) is not continuous on [ 0,∞).
EX 6.14 Using the Comparison Test for an Improper Integral
EX 6.15 Using the Comparison Test for an Improper Integral
Remark
EX 6.16 Using the Comparison Test: A Divergent Integral
