techniques of integration

techniques of integration

1. eliminating square root

   • use trigo identity, and refer graph
   • use when have trigo f(x) in square root, and when trigo f(x) can be simplified

2. improper integral

   • when
   • when denominator can be undefined

3. basic substitution method

   • general exponential f(x)
   • natural log
   • hyperbolic f(x)
     • linear f(x)
       • integrate
     • not linear f(x)
       • use substitution method, then integrate
   • inverse trigo f(x)
     • stewart's table
   • exponential f(x)
     • get in exponential, then integrate
   • trigo f(x)
     • indefinite
     • definite
   • inverse hyperbolic f(x)

4. trigonometric integrals

   • power(even), use half angle formula
   • both power(odd), factorize the smallest power
   • power 1, use identities
   • power is too big, use reduction formula
   • both power(even), use half angle formula
   • power(odd), factorize until either one got power 1
   • power(odd&even), factorize the odd power

5. by parts

   • when cannot use subs method
   • choose u & dv
     • • integrate dv
       • differentiate u (ILATE)
   • tabular integration
     • stop when last eq = ques
     • when differentiate u=0
       • differentiate until get 0, integrate dv, make arrow
     • when differentiate u never get 0

6. partial fraction

   • non repeated linear factors
     • heaviside 'coverup' method, factorize lowest degree, integrate
   • repeated linear factors
     • when denominator repeated
     • find v- by comparing the coefficient
   • improper fraction
     • use long division method, heaviside method, integrate
   • irreducible quad factors
     • when quad factor cannot be factorize
     • heaviside method, integrate

7. trigo substitution

8. trigonometric identities

   • use when have summation of two terms power two, and when have different angle
   • expand first, then apply trigo identity

9. improper fraction

   • when numerator > denominator
   • how?
     • long division method, integrate

10. separating fractions

    • use substitution method, then integrate
    • use when fraction can be separated, and result to a simpler intergrand

11. completing the square

    • use when no substitution available and when have quadratic f(x) at denominator
    • use completing the square, then integrate

12. multiplying by form of 1

    • use when the equation is in 1 term
    • how?
      • conjungate , identity, separate, integrate

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