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### Table of Content:

1. Introduction and Background
1. The Riemann integral
2. The fundamental theorem of calculus
3. Theorems on integration
4. Standard integrals
5. Integration by parts
6. Improper integrals
7. Non-uniqueness of representation
8. Exercises 1
2. The integration of rational functions
1. Improper fractions
2. Linear denominator, Q(x)
4. Cubic denominator, Q(x)
5. Exercises 2
3. The integration of trigonometric functions
1. Simple products
2. Powers of sin and cos
3. Rational functions of sin and cos
4. Exercises 3
4. Part II The integration of ordinary differential equations
5. List of Equations
6. Preface
7. What is a differential equation?
1. The nature and solution of differential equations
2. Classification of ODEs
3. Overview of the equations to be discussed
8. First order ODEs: standard results
1. Separable equations
2. Homogeneous equations
3. The general linear equation
9. First order ODEs: special equations
1. The Bernoulli equation
2. The Clairaut equation
3. The Riccati equation
4. Exact differentials
5. Missing variables
10. Second order ODEs
1. Constant coefficient equations
2. The Euler equation
3. Reduction of order
4. Variation of parameters
5. Finding particular integrals directly
6. Missing variables
7. Exercises 4
11. More general aspects of ODEs
1. Interchanging x andy
2. Singular solutions
3. Uniqueness
4. Exercises 5
6. Index