1 1 Further Integration

• 1.1 Some important information
• 1.0 Chapter 1 contents
• 1.1 Rational functions
• 1.2 Systematic integration of rational functions
• 1.3 Dividing polynomials
• 1.4 The fundamental theorem of algebra
• 1.5 Factors of real polynomials
• 1.6 Factorizing of real polynomials
• 1.7 Continuing Example 1.3
• 1.8 Quadratic factor $q(x)=x^{2}+2bx+c.$
• 1.9 Form of partial fractions
• 1.10 Sketch of proof of Theorem 1.9 (non-examinable)
• 1.11 Continuation of proof
• 1.12 Completing the proof
• 1.13 General remarks
• 1.14 More on Example 1.3
• 1.15 Integrating the partial fractions
• 1.16 Combining terms
• 1.17 Integrating term (b)
• 1.18 Simple examples
• 1.19 Dealing with double roots
• 1.20 Two distinct linear factors
• 1.21 Example involving a quadratic
• 1.22 Example continued
• 1.23 Complete example
• 1.24 Example continued
• 1.25 Summary of integration of rational functions
• 1.26 Tangent substitutions or $\tan\frac{x}{2}$ substitutions
• 1.27 Reducing $\tan\frac{x}{2}$
• 1.28 Reducing sine and cosine integrals
• 1.29 Wright’s integral (1599)
• 1.30 Continuing Wright’s integral (1599)
• 1.31 Integrals associated with conics
• 1.32 Integral associated with the parabola
• 1.33 Integral associated with the circle
• 1.34 Hyperbolic area
• 1.35 Graph of the hyperbola
• 1.36 Hyperbolic triangles
• 1.37 Standard substitutions for integrals involving $\sqrt{\pm a^{2}\pm x^{2}}$.
• 1.38 Substitutions
• 1.39 Useful identities
• 1.40 Remark on substitutions
• 1.41 Improper integrals
• 1.42 Integrals over infinite ranges
• 1.43 Integrals over infinite ranges
• 1.44 Integration using partial fractions
• 1.45 Integration using partial fractions continued
• 1.46 Examples of limits
• 1.47 An integral from the theory of differential equations
• 1.48 An integral from the theory of differential equations
• 1.49 Integrals over the real line
• 1.50 Example of an integral over the real line
• 1.51 Another integral over the real line
• 1.52 Integrals of unbounded functions
• 1.53 Integrals of unbounded functions
• 1.54 Example of an improper integral
• 1.55 Arcsine integral
• 1.56 Example
• 1.57 Comparison test for integrals
• 1.58 Comparison Test.
• 1.59 Laplace transforms
• 1.60 Laplace transforms
• 1.61 Laplace transforms of trig functions
• 1.62 Laplace transforms of hyperbolic functions