Integration Strategy

Integration Strategy

1. Basic Substitution Method

   • Trigonometric Functions
     • ᶴ sin x dx = - cos x + C
       •    ᶴ sin x dx= - cos x + C ᶴ cos x dx = sin x + C ᶴ sec2 x dx = tan x + C ᶴ sec x tan x = sec x + C ᶴ csc2 x dx = - cot x + C ᶴ csc x cot x dx = - csc x + C   
       • ᶴ cos x dx = sin x + C
         • ᶴ sec^2 x dx = tan x + C
           • ᶴ sec x tan x = sec x + C
             • ᶴ csc^2 x dx = - cot x + C
               • ᶴ csc x cot x dx = - csc x + C
     • Trigonometric Identities
       • cos^2 Ө + sin^2 Ө = 1
         • 1 + tan^2 Ө = sec^2 Ө
           • 1 + cot^2 Ө = csc^2 Ө
     • Addition Formula
       • cos (A + B) = cos A cos B – sin A cos B
       • sin (A + B) = sin A cos B + cos A sin B
     • Double Angle Formula
       • cos 2Ө = cos^2 Ө - sin^2 Ө
         • sin 2Ө = 2 sin Ө cos Ө
     • Half-Angle Formula
   • Natural Logarithm
     • ∫ 1/u du = ln |u| + C
   • Exponential Functions
     • ∫ e^u du = e^u + C
   • General Exponential Functions
     • ∫ a^u du = a^u/ln a + C
   • Inverse Trigonometric Functions
   • Hyperbolic Functions
     • ∫ sinh u du = cosh u + C
       • ∫ cosh u du = sinh u + C
         • ∫ sech^2 u du = tanh u + C
           • ∫ csch^2 u du = - coth u + C
             • ∫ sech u tanh u du = - sech u + C
               • ∫ csch u coth u du = - csch u + C
   • Inverse Hyperbolic Functions

2. Completing the Square

   • ax^2 + bx + c = 0
     • x^2 + b/a x + c/a = 0

3. Trigonometric Identities

   • - summation of two terms power two
     • - Different angle
       • ᶴ csc x dx = -ln l csc x + cot x l + C
         • ᶴ sec x dx = ln l sec x + tan x l + C
           • ᶴ tan x dx = -ln l cos x l + C
             • ᶴ cot x dx = ln l sin x l + C

4. Improper Fraction

   • ᶴ polynomial/polynomial
     • (Degree of numerator ≥ Degree of denominator)
       • Long Division Method

5. Separating Functions

   • (a + b)/c = a/c + b/c

6. Multiplying by a Form of 1

7. Eliminating Square Roots

   • Trigonometric Functions win the Square Root
   • the Trigonometric Functions can be Simplified to Eliminate the Square Roots

8. Integration by Parts

   • Either function is not the differential coefficient of the other
     • eg:: ᶴ x^2 ln x dx
       • ᶴ u dv = uv – ᶴ v du

9. Trigonometric Integrals

   • ᶴ sin2 x dx = ½ ᶴ ( 1 – cos 2x) = x/2 – sin2x/4 + C
     • ᶴ cos2 x dx = ½ ᶴ ( 1 + cos 2x) = x/2 + sin2x/4 + C

10. Trigonometric Substitutions

11. Improper Intergral

    • Type 1
    • Type 2

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