Calculus

1. Integration
   1. Concept of an Integral
      1. Indefinite Integrals and Antiderivatives
         1. Understanding antiderivatives
         2. Constant of integration
         3. Properties of indefinite integrals
      2. Definite Integrals and the Area Under a Curve
         1. Concept of signed area
         2. Properties of definite integrals
         3. Absolute area vs. signed area
      3. Notation and Properties
         1. Notation symbols (∫, dx)
         2. Linearity of integrals
         3. Properties of symmetry and periodic functions
   2. Techniques of Integration
      1. Substitution Method
         1. Change of variables
         2. Understanding the substitution process
         3. Reverse chain rule
         4. Common substitutions
      2. Integration by Parts
         1. Derivation from product rule
         2. LIATE rule for choosing u and dv
         3. Applications in various integrals
         4. Repeated integration by parts
      3. Partial Fraction Decomposition
         1. Decomposition of rational functions
         2. Types of partial fractions
         3. Solving systems of equations for coefficients
         4. Integration of decomposed parts
      4. Trigonometric Integrals
         1. Integrating powers of sine and cosine
         2. Trigonometric identities for integration
         3. Special cases and strategies
      5. Trigonometric Substitutions
         1. Substitutions involving sine, cosine, and tangent
         2. Identifying suitable substitutions
         3. Applications in integrals involving radicals
      6. Improper Integrals
         1. Definition and examples
         2. Convergence and divergence
         3. Techniques for evaluating improper integrals
   3. Applications of Integrals
      1. Area Between Curves
         1. Setting up integrals for areas
         2. Horizontal and vertical strips
         3. Finding intersection points
      2. Volume of Solids of Revolution
         1. Disk method
         2. Washer method
         3. Cylindrical shell method
         4. Volume of known cross-sections
      3. Work and Energy Problems
         1. Calculating work done by a force
         2. Application to lifting and springs
         3. Energy applications in physics
      4. Differential Equations
         1. Solving basic separable differential equations
         2. Application in modeling growth and decay
         3. First-order linear differential equations
      5. Average Value of a Function
         1. Definition and calculation
         2. Interpretation in real-world scenarios
         3. Applications in economics and life sciences