INTEGRATION

INTEGRATION

1. Basic substituition

   • Indefinite Integral

2. Completing the Square

   • no substituition
   • Quadratic cannot factorize
   • Quadratic in square root

3. Trigonometric

   • Basic Identity
   • Integration
   • Addition Formula
   • (1) use summation of two terms the power of two.(2) different angle
   • Half angle

4. Improper Fractions

   • numerator>denominator
     • use long division

5. Eliminating Square Roots

   • Trigonometry function in square root
   • trigonometric function can be simplify to

6. Separating Fraction

   • to get simpler intergrand
   • Example:

7. Multiplying by a Form of 1

   • multiply the integrand by some term divide by itself and form 1 to simplify the integrand

8. Integrating by Parts

   • • U easy differentiate
       • Follow IILATE Rule
     • v must easy integrtaed
   • Type
     • Basic
     • Tubular:1) differentiate to 0 2) use table
     • Repeated Linear Factorp0wer bracktet ( )^n
       • Method
         • 1. Find A, B, C
         • 2. Integrate Normally
     • Irreducible bracket (inside bracket cannot factorize) same method like repeated linear factor
     • Improper fraction
       • Method 1)simplify by long division 2) used repeated linear factor if denominator power bracket

9. Integration of Rational Functions by Partial Fractions

   • Rules:
     • 1) numerator< denominator
       • if not uselong division
     • 2) Factorize denominator
     • 3) Linear Factor
     • Example:

10. Improper Integrals

11. Trigonometric Integrals & Substitutions

    • iMPORTANT fORMULAE
    • How to Integrate
      • Release one of the factors sin x from the power
    • Products of sines and cosines
    • Reduction Formula
    • Integrals of the form
    • Products of Power of Sines and cosines
      • Case 1 = If m is odd, we write m as 2k+1 and use the identity sin^2 x = 1-cos^2 x
        • Case 2 = If m is even and n is odd, we write n as 2k+1 and use the identity cos^2 = 1-sin^2 x
          • Case 3 = If both m and n are even, we substitute a) sin^2 x = 1-cos^2x /2 and b) 1+cos^2x/2

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