Review Of Formulas And Techniques Integration Table

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