Integration strategies

Integration strategies

1. 1. Simplify

   • Simplification and Modification
     • With : Basic trig and algebraic identities
       • eg: sin ^2 x + cos ^2 x =1
         • Purpose : To deal with a less complicated integration

2. 2. Check for substitution

   • For example: by taking u = 2x + 3

3. 3. Identify integral type

   • - Rational function, Exponential Function, Logarithm, Trigonometric function, Inverse Trigo, Hyperbolic Function, Inverse Hyperbolic, Quadratic Functions

4. 4. Relating and Manipulating

   • Example:
     • Substitute x = sin θ then dx = cos θ dθ.

5. 5. Need for Multiple techniques?

   • 11 Techniques:
     • 2) Completing the square
       • When to use?
         • When no substitution available, quadratic function at denominator, (and cannot be factorized) or it is in the square root
     • 3) Trigonometric Identities
     • 4) Improper Fraction
       • Degree of numerator larger or equal to dEgree of denominator
         • Use Long Division Method
     • 5) Integration of Rational Functions by Partial Fractions
       • The degree of numerator less than degree of denominator
         • District linear factors (also heaviside "cover up" method)
         • Irreducible quadratic factor
         • Repeated irreducible factor
         • Repeated linear factors
         • Improper fraction
     • 6) Multiplying by a form of 1
     • 7) Eliminating square roots
       • When to use?
         • Trigo function in square root and it can be simplified (using trigo identities to a squared trigo form so that sqaure root can be eliminated)
     • 8) Integration by parts
       • Either function not differentiation, coefficient of the other
         • Hence, use
           • eg.
           • ILATE rule
     • 9) Separating Fraction
     • 10) (a) Trigonometric Integral
       • higher powers of sin x and cos x (reduction formula)
         • 6 reduction formula
       • Product of sines and cosines
       • Product of powers of sines and cosines
     • 11) Improper Integrals
     • 1) Basic Substitution Methods
     • 10) (b) Trigonometric Substitutions
       • eg.

6. 6) Try Again

   • If things do not work out, go through the process and try again. but, try a different technique this time.

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