TECHNIQUES OF INTEGRATION

TECHNIQUES OF INTEGRATION

1. Basic Substitution Method

2. Completing The Square

   • WHEN?
     • no obvious substitution to make
     • The denominator does not factor with rational coefficients
   • HOW?
     • rewrite any quadratic polynomial ax2+bx+x in form a(x+k1)^2+k2
     • 2x^2−12x+26 =2[x^2−6x+13] =2[(x^2−6x+9)+4] =2[(x−3)^2+4]

3. Trigonometric Identities

4. Improper Fraction

5. Separating Fraction

6. Multiplying by A Form of 1

7. Eliminating Square Roots

8. Integration by Parts

   • WHEN?
     • to integrate a product where either function is not the differential coefficient of the other
   • HOW?
     • Normal
       • 1) express the equation in the terms of u and v
       • 2) use formula to integrate the equation
       • Formula: ∫ u dv = uv - ∫ v du
     • Tabular
       • 1) express the equation in the terms of u and v
       • 2) use formula to integrate the equation
   • EXAMPLE
     • Normal
     • Tabular

9. Integration of Rational Functions by Partial Fractions

   • WHEN?
     • numerator is not the differential coefficient of the denominator
   • HOW?
     • 1) The numerator of the given function must be of lower degree than that of the denominator. If it is not, then first divide out by long division
     • 2) Factorise the denominator into its prime factors
     • 3) Express the function in terms of its partial fractions

10. Trigonometric Integrals

11. Trigonometric Substitutions

12. Improper Integrals

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