INTEGRATION TECHNIQUES

INTEGRATION TECHNIQUES

1. Basic Substitution Method

   • modify to get the form
     • general exponential
     • exponential fn.
     • natural log
   • how to choose u: inside bracket or inside square root
   • for nonlinear fn.
   • indefinite integral
     • sub u back after integrate
   • definite integral
     • change lower and upper limit after choosing u then subs new limit to get final answer

2. Completing the Square

   • When?
     • Cannot use Substitution
     • Quad. fn at denominator
       • in square root
       • cannot factorize

3. Trigonometric Identities

   • When?
     • summation of 2 terms power two
     • different angle

4. Improper Integrals

   • • It will become undefined

5. Trigonometric Substitutions

6. Integration by Parts

   • either fn. is not the differential coefficient of other
     • How to identify?
       • ***
         • use ILATE
       • Expression of u is easy to differentiate while dv is easy to integrate
   • Tabular integration
     • differentiate repeatedly until become zero
     • special case
       • stop until get - sign

7. Separating Fractions

   • to get simpler integrand

8. Multiplying by form of 1

   • multiply the integrand by some term divided by itself

9. Eliminating Square Root

   • When?
     • trigonometric fn. in square root
     • trigo fn. can be simplify
       • HINT!! - use trigo identities -refer graph (check sign) -manipulate to get power 2

10. Trigonometric Integrals

    • • same way as we found the integral of sin^3 x
    • • Half - Angle Formulae
    • • use reduction formula
    • • release one of the factors sin x from the power and convert the remaining sin^2 x into (1-cos^2 x)
    • product of powers of sines and cosines
      • • divide appropriate subs into 3 cases according to mand n being odd or even
    • product of sines and cosines
      • • use trigo identity
    • • subs t = tan x
    • • subs t = tan x/2

11. Improper Fraction

    • How to solve?
      • use long division method

12. Integration of Rational Functions by Partial Fraction

    • by Partial Fraction
    • nonrepeated (heaviside cover up method)
    • repeated
    • irreducable quad. fn.
      • quadratic fn which cannot be factorize
    • improper fraction
      • use long division method
    • transcedental

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