Techniques of Integration

Techniques of Integration

1. Separating fractions (when the fraction can be separated)

   • 1. Separate the function into 2 eq. with same denominator.
   • 2. Simplify then integrate each term

2. Improper integral

   • 1. sketch the graph
   • 2. change into limit
   • 3. integrate the equation
   • 4. solve the limit

3. Trigonometric integrals

   • when the trigo fn is to the power of an odd no.
     • use half angle formula, then integrate using basic subs. method
   • when trigo fn is to the power of an odd no.
     • 1. release one of the factor; convert the remaining using identities
     • 2. integrate each term
   • Reduction formula
     • to integrate sin^n x & con^n x
   • t-substitution (t = tan x)
     • 1. change the eq. in terms of t
     • 2. integrate the fn
     • 3. convert back the t into tan x
   • product of powers of sines & cosines
     • both odd
     • one odd one even
     • both even

4. Multiplying by a form of 1

   • 1. multiply by a form of 1 or its conjugate
   • 2. simplify the expression

5. Integration by parts

   • tabular integration
     • 1. differentiate until it becomes 0
     • 2. combine the product of fn
   • in the form of
     • 1. identity u and dv
       • use ILATE
     • 2. substitute into the form

6. Improve fraction (when degree of numerator is >/= degree of denominator

   • 1. use long division method to simplify the equation
   • 2. integrate the simplified eq. as usual
   • basic substitution method
     • 1. Identity the u
     • 2. Differentiate u and change the question to the form of u
     • 3. Integrate as usual

7. Completing the square, CSM (when quadratic function at denominator)

   • 1. Simplify denominator using CSM
   • 2. Integrate the equation
   • *answer will usually be in the form of inverse trigo

8. Trigonometric identities (when there is summation of 2 terms: 2 @diffrent angle)

   • 1. expand the equation
   • 2. apply the trigonometric identities
   • 3. make sure every term can be integrated
   • 4. integrate each term as usual

9. Eliminating square roots (when trigo function is in the square root)

   • 1. simplify to a squared trigonometric form
   • 2. eliminate the square root
   • 3. integrate as usual

10. Integration by partial fractions

    • Non-repeated linear factors
      • 1. simplify the denominator
      • 2. separate the denominator
      • 3. find A and B
        • use Heaviside "cover up" method
    • Repeated linear factors
    • Irreducible quadratic factors
    • Improper fraction

11. Trigonometric substitution

    • 1. convert the variable in terms of theta
    • 2. integrate as usual
    • 3. change the theta back into variable form

Images and files in the map