STRATEGIES FOR WAR IN INTEGRATION

STRATEGIES FOR WAR IN INTEGRATION

1. Completing The Square

   • 1. Substitution cannot be used
   • 2. There is denominator : quadratic function and that quadratic function
     • Cannot Factorize
     • In square root form

2. Trigonometric Identities

   • 1. When substitution does not work
   • 2. (1 term + 1 term)^2, expand first!
   • 3. when having different angles

3. Improper Fraction

   • 2. Degree of numerator >/= degree of denominator
     • Long Division Method
   • 1. Substitution does not work

4. Separating Fractions

   • 1. Substitution does not work
   • 2. Fraction can be seperated
     • Result to a simpler integrand

5. Multiplying by Form of 1

   • 1. Multiply expression which divided by itself to simply equation

6. Eliminating Square Roots

   • 1. Substitution does not work
   • 2. Use this when trigo function is in square root
   • Can be simplified to a squared trigonometric form to eliminate the square root

7. Integration by Parts

   • 1. Substitution does not work
   • 3. There is a product form
     • Use ILATE rules (from E to I)
       • I: Inverse Trigonometric
       • L : Logarithm
       • A : Algebraic
       • T: Trigonometric
       • E : Exponential
   • 2. Can be differentiated until become ZERO
   • If integrand a polynomial X trigonometric function/exponential/logarithm

8. Rational Functions

   • 1. Basic Substitution does not work
   • 2. Degree of numerator < degree of denominator
   • 3. Factorize denominator ----> prime factor

9. Trigonometric

   • Integrals
     • 1. Substitution cannot be used
     • 2. We need to know the trigo identities
     • 3. If we have higher power of sin x and cos x, so use reduction formula
     • 4. Then settle the question by using integration by part
   • Substitution
     • 1. Replace the variable of integration by a trigonometric function

10. Improper Integrals

    • Use limit
      • Type I
      • Type II

11. Basic Substitution Method

    • When there is two functions in integral
    • When differentiate one funtion, the expression can be simplified

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