Lecture Notes on Calculus and Integration

Key Concepts Covered

Basic Integration Formulas and Techniques

• Integration of basic trigonometric functions:
  • [ \int \sec^2(x)dx = \tan(x) + C ]
  • [ \int \csc^2(x)dx = -\cot(x) + C ]
  • [ \int \sec(x)\tan(x)dx = \sec(x) + C ]
  • [ \int \csc(x)\cot(x)dx = -\csc(x) + C ]
• Standard integral forms involving trigonometric identities:
  • [ \int \sin^2(x)dx ]
  • [ \int \cos^2(x)dx ]
  • Rewriting sin and cos squared integrals using half-angle identities: [ \cos^2(x) = \frac{1+\cos(2x)}{2} \gram \sin^2(x) = \frac{1-\cos(2x)}{2} ]
• Integration by substitution:
  • Example of [ \int \frac{dx}{\sqrt{1-x^2}} = \sin^{-1}(x) + C ]
• Handling integrals of the form ( \int \frac{dx}{a^2-x^2} ) using arcsine transformations.
• Integration using inverse trigonometric functions: [ \int \frac{dx}{\sqrt{a^2-x^2}} = \frac{\sin^{-1}(x/a)}{a} + C ]
• Evaluating integrals involving logarithmic functions:
  • [ \int \frac{dx}{x} = \ln|x| + C ]

Complex Integration Techniques

• Integration by parts: [ \int u dv = uv - \int v du ]
  • Application examples including ( \int x e^x dx )
• Solving integrals involving products of polynomials and exponentials: [ \int x^n e^x dx ]
• Handling integrals with multiple functions:
  • Using the method of partial fractions to decompose complex rational integrals.
  • [ f(x) = \frac{A}{(x-a)} + \frac{B}{(x-b)} ]

Applications of Integration

• Calculating areas under curves using definite integrals:
  • Setting up integrals to find the area between curves.
  • Example: calculating the area between a parabola and a linear function.
  • Finding limits to integrate over regions: [ \int_a^b f(x) dx ]

Special Cases and Advanced Techniques

• Integrating functions involving roots and squares:
  • [ \int \sqrt{a^2 - x^2} dx ] transformations using trigonometric identities (e.g., setting ( x = a \sin(\theta) )).
• Evaluating definite integrals using symmetry and known geometric shapes.
• Integration of functions involving parametric equations and transformations:
  • Example: converting polar coordinates to Cartesian coordinates and integrating.

Example Integrals and Solutions

• Detailed examples of evaluated integrals:
  • [ \int_{0}^{1} \sqrt{1 - x^2} dx ] using substitution and known integral formulas.
  • Using polynomial long divisions to simplify integrals before evaluating.
• Calculating definite integrals for real-world applications:
  • Examples involving physical applications like calculating work done or areas under physical curves.

Summary

• Reviewed basic and advanced integration techniques, including substitution, parts, and trigonometric integrals.
• Explored applications of definite integrals in calculating areas between curves and other practical uses.
• Detailed walkthroughs of complex integral evaluations using various techniques.

Note: Practice problems and additional examples are crucial for mastering these concepts. Review steps for each type of integral and verify solutions using differentiation.