Integration Strategy

Integration Strategy

1. Basic Substitution

   • Choose 'u'
     • polynomial fn
     • function inside √
     • derivative of 'u' is in the equation
     • for natural log, get √1/x
     • for exponential fn, get e^x, e^fx, f'(x)
     • general exp fn: u = power of variable (a)
   • find du
   • std form
     • ∫u du

2. completing square

   • quadratic fn
     • 2 terms: 2 variables
     • 3 terms: 2 variables 1 const

3. trigonometric identities

   • different angle
   • Σ(a+b)^2

4. improper fraction

   • Degree of numerator ≥ degree of denominator
     • long division method

5. separating fraction

6. multiply by a form of 1

7. eliminating √ (trigo fn)

   • using trigo identity

8. Trigo integral

   • ∫sin x and ∫cos x with power > 2
     • if power is odd, separate to even and odd
     • if power is even, manipulate to power of 2
     • use reduction formula
   • Product of power of sines and cosines
     • ∫sin^m x ∫cos^n x
       • Case 1 : m is odd
         • Substitute
       • Case 2: m is even and n is odd
         • split the odd
           • substitute
       • Case 3: both m and n are even
         • use half angle
           • substitute
     • ∫sin 4x. cos 2x
       • Use identity 2 sin A cos B = sin (A+B) + sin (A-B)
   • Integrals in the form of
     • • Substitute t = tan x
     • • substitute t = tan x/2

9. Integration by parts

   • ∫ u dv
     • dv is easy to integrate
     • u easy to differentiate
       • ILATE rule
     • tabular integration
       • differentiate until get zero

10. rational function by partial fraction

    • use Long Divison if improper
      • factorize denominator
        • choose case
          • non-repeated linear factor
          • repeated linear factor
          • irreducible quadratic factor
          • transedental function

11. Trigonometric Substitution

    • Transforming integrals
    • Substitute x with values of Ѳ
      • change x to Ѳ term
        • integrate and change back to x term

12. Improper integral

    • Type 1
      • limits has infinity
    • Type 2
      • integrand becomes infinite

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