Integral Calculus - Integration of a given function

  • Definition of integration

  • Anti-derivative and indefinite integrals

  • Notation for indefinite integrals

  • Examples of indefinite integrals

  • Properties of indefinite integrals

  • Meaning of definite integrals

  • Geometrical interpretation of definite integrals

  • Notation for definite integrals

  • Properties of definite integrals

  • Examples of definite integrals

  • Fundamental Theorem of Calculus - Part 1

  • Statement of the theorem

  • Interpretation of the theorem

  • Application of the theorem

  • Examples illustrating the theorem

  • Fundamental Theorem of Calculus - Part 2

  • Statement of the theorem

  • Interpretation of the theorem

  • Application of the theorem

  • Examples illustrating the theorem

  • Integration techniques - Substitution method

  • Procedure for solving integrals using substitution

  • Examples illustrating the substitution method

  • Integration by parts method

  • Procedure for solving integrals using integration by parts

  • Examples illustrating the integration by parts method

  • Integration of trigonometric functions

  • Integration of exponential functions

  • Integration of logarithmic functions

  • Examples illustrating the integration of various functions

  • Integration of rational functions

  • Procedure for solving integrals involving rational functions

  • Examples illustrating the integration of rational functions

  • Integration of trigonometric substitutions

  • Procedure for solving integrals using trigonometric substitutions

  • Examples illustrating the trigonometric substitutions method

  • Integration of partial fractions

  • Technique for solving integrals using partial fractions

  • Examples illustrating the partial fractions method

  • Integration of improper integrals

  • Definition of improper integrals

  • Classification of improper integrals

  • Evaluation of improper integrals

  • Examples illustrating the improper integrals

  • Applications of integration in real life

  • Calculation of area using integration

  • Finding the area under a curve

  • Application of integration in physics

  • Application of integration in economics

  • Summary and review of integral calculus Here are slides 11 to 20 on the topic “Integral Calculus - Integration of a given function”:

Slide 11:

  • Integration techniques - Substitution method
  • Procedure for solving integrals using substitution
  • Examples illustrating the substitution method
  • Integration by parts method
  • Procedure for solving integrals using integration by parts

Slide 12:

  • Examples illustrating the integration by parts method
  • Integration of trigonometric functions
  • Integration of exponential functions
  • Integration of logarithmic functions
  • Examples illustrating the integration of various functions

Slide 13:

  • Integration of rational functions
  • Procedure for solving integrals involving rational functions
  • Examples illustrating the integration of rational functions
  • Integration of trigonometric substitutions
  • Procedure for solving integrals using trigonometric substitutions

Slide 14:

  • Examples illustrating the trigonometric substitutions method
  • Integration of partial fractions
  • Technique for solving integrals using partial fractions
  • Examples illustrating the partial fractions method
  • Integration of improper integrals

Slide 15:

  • Definition of improper integrals
  • Classification of improper integrals
  • Evaluation of improper integrals
  • Examples illustrating the improper integrals
  • Applications of integration in real life

Slide 16:

  • Calculation of area using integration
  • Finding the area under a curve
  • Application of integration in physics
  • Application of integration in economics
  • Summary and review of integral calculus

Slide 17:

  • Definition of integration
  • Anti-derivative and indefinite integrals
  • Notation for indefinite integrals
  • Examples of indefinite integrals
  • Properties of indefinite integrals

Slide 18:

  • Meaning of definite integrals
  • Geometrical interpretation of definite integrals
  • Notation for definite integrals
  • Properties of definite integrals
  • Examples of definite integrals

Slide 19:

  • Fundamental Theorem of Calculus - Part 1
  • Statement of the theorem
  • Interpretation of the theorem
  • Application of the theorem
  • Examples illustrating the theorem

Slide 20:

  • Fundamental Theorem of Calculus - Part 2
  • Statement of the theorem
  • Interpretation of the theorem
  • Application of the theorem
  • Examples illustrating the theorem

Slide 21:

  • Integration by partial fractions
    • Decomposing a rational function into partial fractions
    • Procedure for solving integrals involving partial fractions
    • Examples illustrating the integration by partial fractions method
  • Integration of trigonometric functions
    • Trigonometric identities used in integration
    • Evaluation of integrals involving trigonometric functions
    • Examples illustrating the integration of trigonometric functions

Slide 22:

  • Integration of exponential functions
    • Properties of exponential functions
    • Evaluation of integrals involving exponential functions
    • Examples illustrating the integration of exponential functions
  • Integration of logarithmic functions
    • Properties of logarithmic functions
    • Evaluation of integrals involving logarithmic functions
    • Examples illustrating the integration of logarithmic functions

Slide 23:

  • Integration of inverse trigonometric functions
    • Properties of inverse trigonometric functions
    • Evaluation of integrals involving inverse trigonometric functions
    • Examples illustrating the integration of inverse trigonometric functions
  • Integration of hyperbolic functions
    • Properties of hyperbolic functions
    • Evaluation of integrals involving hyperbolic functions
    • Examples illustrating the integration of hyperbolic functions

Slide 24:

  • Integration of products of trigonometric functions
    • Use of trigonometric identities in integration
    • Evaluation of integrals involving products of trigonometric functions
    • Examples illustrating the integration of products of trigonometric functions
  • Integration of products of exponential and logarithmic functions
    • Use of properties of exponential and logarithmic functions in integration
    • Evaluation of integrals involving products of exponential and logarithmic functions
    • Examples illustrating the integration of products of exponential and logarithmic functions

Slide 25:

  • Integration of functions with square roots
    • Evaluation of integrals involving square roots
    • Techniques for simplifying integrands with square roots
    • Examples illustrating the integration of functions with square roots
  • Integration of functions with rational exponents
    • Conversion of rational exponents to fractional powers
    • Evaluation of integrals involving rational exponents
    • Examples illustrating the integration of functions with rational exponents

Slide 26:

  • Integration of functions involving absolute values
    • Evaluation of integrals involving absolute values
    • Cases for splitting the integral based on the sign of the function
    • Examples illustrating the integration of functions involving absolute values
  • Integration of functions with logarithmic derivatives
    • Evaluation of integrals involving logarithmic derivatives
    • Techniques for simplifying integrands with logarithmic derivatives
    • Examples illustrating the integration of functions with logarithmic derivatives

Slide 27:

  • Integration of parametric equations
    • Conversion of parametric equations to Cartesian form
    • Evaluation of integrals in terms of the parameter
    • Examples illustrating the integration of parametric equations
  • Integration using numerical methods
    • Trapezoidal rule for numerical integration
    • Simpson’s rule for numerical integration
    • Examples illustrating the use of numerical methods for integration

Slide 28:

  • Integration using series
    • Power series representation of functions
    • Evaluation of integrals using series expansions
    • Examples illustrating the use of series for integration
  • Application of integration in geometry
    • Calculation of arc length
    • Calculation of surface area
    • Examples illustrating the application of integration in geometry

Slide 29:

  • Application of integration in calculus
    • Calculation of limits using integration
    • Calculation of derivatives using integration
    • Examples illustrating the application of integration in calculus
  • Application of integration in engineering
    • Calculation of work done
    • Calculation of fluid flow
    • Examples illustrating the application of integration in engineering

Slide 30:

  • Review of integration techniques
    • Summary of integration methods
    • Tips for choosing the appropriate method
    • Examples illustrating the use of different integration techniques
  • Practice problems and exercises
    • Set of problems for further practice
    • Solutions and step-by-step explanations
  • Question and answer session
    • Addressing doubts and queries from students
    • Clarifying concepts related to integration