Calculus

  1. Techniques of Integration
    1. Substitution Method
      1. u-Substitution for Indefinite Integrals
        1. u-Substitution for Definite Integrals
          1. Trigonometric Substitutions
            1. Substitutions for √(a²-x²)
              1. Substitutions for √(a²+x²)
                1. Substitutions for √(x²-a²)
                2. Advanced Substitution Techniques
                3. Integration by Parts
                  1. The Integration by Parts Formula
                    1. Strategy for Choosing u and dv
                      1. Repeated Integration by Parts
                        1. Tabular Integration
                          1. Integration by Parts for Definite Integrals
                          2. Trigonometric Integrals
                            1. Integrals of Powers of Sine and Cosine
                              1. Integrals of Powers of Tangent and Secant
                                1. Integrals of Products of Trigonometric Functions
                                  1. Reduction Formulas
                                  2. Partial Fraction Decomposition
                                    1. Rational Functions and Proper Fractions
                                      1. Linear Factors
                                        1. Distinct Linear Factors
                                          1. Repeated Linear Factors
                                          2. Quadratic Factors
                                            1. Distinct Irreducible Quadratic Factors
                                              1. Repeated Irreducible Quadratic Factors
                                              2. Integration After Decomposition
                                              3. Numerical Integration
                                                1. Midpoint Rule
                                                  1. Formula and Error Analysis
                                                  2. Trapezoidal Rule
                                                    1. Formula and Error Analysis
                                                    2. Simpson's Rule
                                                      1. Formula and Error Analysis
                                                      2. Comparison of Methods
                                                      3. Improper Integrals
                                                        1. Type I: Infinite Intervals
                                                          1. Integrals from a to ∞
                                                            1. Integrals from -∞ to a
                                                              1. Integrals from -∞ to ∞
                                                              2. Type II: Discontinuous Integrands
                                                                1. Integrals with Discontinuities at Endpoints
                                                                  1. Integrals with Interior Discontinuities
                                                                  2. Convergence Tests
                                                                    1. Comparison Test
                                                                      1. Limit Comparison Test

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                                                                  6. The Definite Integral

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                                                                  8. Applications of Integration