Techniques of Integration

Techniques of Integration

1. basic substitution method

   • 1) identify the U
   • 2) differentiate U and change the question to the form of U
   • 3) integrate as usual

2. completing the square,CSM (when quadratic function at denominator)

   • 1) simplify denominator using CSM
   • 2) integrate the equation
   • *answer usually in the form of inverse trigo.

3. trigonometric identities (when there is summation of 2 terms ^2 @ different angle)

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

4. improper fraction (when degree of numerator>/= degree of denominator)

   • 1) use long division method to simplify the equation
   • 2) integrate the simplified eq. as usual

5. separating fractions (when the fraction can be separated)

   • 1) separate the function into 2 eq. with same denominator.
   • 2) simplify then integrate each term

6. multiplying by a form of 1

   • 1) multiply by a form of 1 or its conjugate
   • 2) simplify the expression
   • 3) integrate the expression as usual.

7. 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

8. integration by parts

   • in the form of
     • 1) identify u and dv
       • use ILATE
     • 2) substitute into the form
   • tabular integration
     • 1) diff u until becomes 0
     • 2) combine the product of fn

9. integration by partial fractions

   • non-repeated linear factors
     • 1) simplify the deniminator
     • 2) separate the denominator
     • 3) find A and B
       • use Heaviside "cover up" method
   • repeated linear factors
   • irreducible quadratic factors
   • improper fraction

10. trigonometric integrals

    • when the trigo fn is to the power of an even 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 and cos^n x
    • product of powers of sines and cosines.
      • both odd
      • one odd one even
      • both even
    • 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

11. trigonometric substitution

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

12. improper integral

    • 1) sketch the graph
    • 2) change into limit
    • 3) integrate the equation
      • By: Hamidy and Anas (set7)
    • 4) solve the limit.

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