1_11 Strategies In Integrations

1_11 Strategies In Integrations

1. Basic Substitution

   • 1) Take function that have high power to be as U and differentiate it
   • 2) Replace integral with variable U and du, the integrate it
   • 3) After integrate, replace back U with the original function

2. Completing The Square

   • Basic substitution is not available
   • To get 1 constant and 1 variable

3. Trigonometric Identities

   • sin2x + cos2x = 1
     • 1 + tan2x = sec2x
     • 1 + cot2x = csc2x
   • Addition Formulas
     • cos (A+B) = cos A cos B - sin A sin B
     • sin (A+B) = sin A cos B + cos A sin B
   • Double-Angle Formulas
     • sin 2x = 2 sin x cos x
     • cos 2x = cos2x - sin2x
   • Half-Angle Formulas
     • cos2x = (1 + cos2x)/2
     • sin2x = (1 - cos2x)/2

4. Improper Fraction

   • Use long division for polynomials
   • Basic substitution is not available

5. Separating Fractions

   • Applicable when the fractions can be separated
   • To get simpler integrand

6. Multiplying By A Form of 1

   • Used to multiply the integral by some term divided by itself
   • To get simpler integrand
   • Basic substitution, completing the square, improper fraction, and separating function are not available

7. Eliminating Square Roots

   • Used when have a trigonometric function in the square root
   • Used when trigonometric functions can be simplified by using trigonometric identities to a squared trigonometric form
   • Sketch the graph to solve the absolute integrand

8. Integration By Parts

   • 1) Integral u dv = uv - integral v du
   • A right choose of u by using ILATE RULE while dv is easy to integrate
     • I : INVERSE TRIGO / INVERSE HYPERBOLIC
     • L : LOGARITHMIC / GENERAL LOGARITHMIC
     • A : ALGEBRAIC
     • T: TRIGONOMETRIC / HYPERBOLIC
     • E : EXPONENTIAL / GENERAL EXPONENTIAL
   • Used when Basic Substitution, Completing the Square, Trigonometric Identities, Improper Fraction, Separating Fractions, Multiplying by a Form of 1, and Eliminating Square Roots do not work
   • 2) Tabular Integration ONLY FOR :
     • Integral ALGEBRAIC . TRIGONOMETRIC dx
     • Integral ALGEBRAIC . EXPONENTIAL dx