TECHNIQUES OF INTEGRATION

TECHNIQUES OF INTEGRATION

1. BASIC SUBSTITUTION METHOD

2. COMPLETING THE SQUARE

   • Rules
     • no substitution available
     • have quadratic function at denominator
       • quadratic function cannot be factorize
       • quadratic function is in the square root

3. TRIGONOMETRIC IDENTITIES

   • Rules
     • the summation of two terms power of two
     • have different angle

4. IMPROPER FRACTIONS

   • Rules
     • the degree of numerator ≥ the degree of denominator
   • Method
     • Long Division Method

5. SEPARATING FACTORS

   • Rules
     • the fraction can be separated
     • result to a simpler integrand
   • Method
     • Separate the numerator

6. MULTIPLYING BY A FORM OF 1

   • Rules
     • Can be simplified to an expression that can be integrated
   • Method
     • multiply the expression by some term divided by itself

7. ELIMINATING SQUARE ROOTS

   • Rules
     • have a trigonometric function in the square root
     • the trigonometric function can be simplified
   • Method
     • simplify by using trigonometric identities
       • eliminate the square root

8. INTEGRATION BY PARTS

   • Rules
     • when the 7 techniques do not work
     • function is not the differential coefficient of the other
   • Method

9. INTEGRATION OF RATIONAL FUNCTIONS BY PARTIAL FRACTIONS

   • Rules
     • the numerator of the given function must be of lower degree than that of the denominator
     • a linear factor (ax + b) gives a partial fraction of the form A/(ax+b)
     • Factors (ax+b)^2 give partial fraction of the form A/(ax+b) + B/(ax+b)^2
     • Factors (ax+b)^3 give partial fraction of the form A/(ax+b) + B/(ax+b)^2 + C/(ax+b)^3
     • A quadratic factor (ax^2 + bx + c) give partial fraction of the form Ax+B/(ax^2+bx+c)
   • Methods
     • Factorize the denominator into its prime factors
     • Heaviside "cover up" method - only for non-repeated linear factor
       • use long division if the numerator is at lower degree than that of the denominator

10. TRIGONOMETRIC INTEGRALS

11. TRIGONOMETRIC SUBSTITUTIONS

    • Rules
      • the variable of integration is replaced by a trigonometric function
    • Methods
      • √(a^(2 )-x^2 ) : use the substitution x=a sin θ, for - π/2 ≤ π/2 or x=a cos θ, for 0 ≤ θ ≤ π
      • √(a^(2 )+x^2 ) : use the substitution x=a tan θ, for - π/2 ≤ π/2
      • √(x^(2 )-a^2 ) : use the substitution x=a sec θ, for 0 ≤ θ ≤ π, θ ≠ π/2

12. IMPROPER INTEGRALS