Cal2 Chapter1

Chapter 1: Integration Techniques

Chapter Outline

  • Integration by Substitution

  • Integration by Parts

  • Integration of Rational Functions Using Partial Fractions

  • Trigonometric Techniques of Integration

  • Integrals involving logarithmic, exponential and hyperbolic functions

  • Improper Integrals

Basic Integration Formulas

  • General Formulas:

    • [ \int k , dx = kx + C ]

    • [ \int x^n , dx = \frac{x^{n+1}}{n+1} + C ] (for ( n
      eq -1 ))

    • [ \int e^x , dx = e^x + C ]

    • [ \int \sin x , dx = -\cos x + C ]

    • [ \int \cos x , dx = \sin x + C ]

    • [ \int \tan x , dx = \ln |\sec x| + C ]

    • [ \int \sec x , dx = \ln |\sec x + \tan x| + C ]

Integration by Substitution

  • Rule: If ( u = g(x) ), then:[ \int f(g(x))g'(x) , dx = \int f(u) , du ]

  • Steps:

    1. Define ( u = g(x) ).

    2. Replace ( g(x) ) with ( u ) in the integrand and ( g'(x)dx ) with ( du ).

    3. Integrate with respect to ( u ).

    4. Substitute back to express the result in terms of ( x ).

Example of Integration by Substitution

  • Procedure:

    • Let ( u = x^2 + 2x + 3 )

      • Calculate ( du = (2x + 2)dx )

      • Perform integration and revert ( u ) back to terms of ( x ).

Integration of Natural Logarithm Function

  • Substitution:

    • Use substitutions like ( u = 3 + \cos x ) or ( u = x^2 - 3 ).

    • Apply logarithmic integration rules.

Tabular Integration

  • Situation: For repeated integration by parts.

  • Method: List derivatives of ( f(x) ) and integrals of ( g(x) ) in a table.

Trigonometric Techniques of Integration

  • Three Cases:

    1. Odd power of sine: Use ( \sin^2 x = 1 - \cos^2 x ) and substitute.

    2. Odd power of cosine: Use ( \cos^2 x = 1 - \sin^2 x ) for substitution.

    3. Even powers of sine and cosine: Use half-angle formulas to simplify.

Improper Integrals

  • Type I: Infinite limits of integration.

    • [ \int_a^\infty f(x) , dx = \lim_{b \to \infty}\int_a^b f(x) , dx ]

  • Type II: Discontinuous integrals.

    • [ \int_a^b f(x) , dx = \lim_{c \to a+}\int_c^b f(x) , dx ] if ( f(x) ) is discontinuous at ( a ).

Derivatives and Integrals of Hyperbolic Functions

  • Key Functions:

    • ( \sinh u ) and ( \cosh u ) have similar integration rules.

Examples and Evaluations

  • Real-World Applications:Use integration techniques to evaluate areas under curves, solve physical problems, etc.