INTEGRATION

INTEGRATION

1. (c). BASIC SUBSTITUION

   • 1. Indefinite Integral
   • 2. Definite Integral

2. (b). REVISION

   • 1. Indefinite Integral
     • : (x^n+1)/n+1 +c n≠1
   • 2. Definite Integral
     • (ax+b)^n+1 / a(n+1) + c

3. (d). TRIGONOMETRIC FUNCTION

   • Identities
   • 1. Simple Integral
   • 2. Basic Substitution Method-Indefinite Intgral
   • 3. Definite Integral
   • Derivatives
   • Integral

4. (e). NATURAL LOGARITHM

   • Derives
   • Integrate

5. (f). EXPONENTIAL FUNCTION

   • Derivative
   • Intergrate
     • ∫ e^u du = e^u + c

6. (g). THE GENERAL EXPONENTIAL FUNCTION a^x

   • Derive
     • d/dx a^u = a^u ln a du/dx
   • Integrate
   • For a>0 and a^x = e^x ln a

7. (h). INTEGRATION: INVERSE TRIGONOMETRIC FUNCTION

   • Integration
   • Derivatives

8. (i). HYPERBOLIC FUNCTION

   • Derivatives
   • Integral

9. (j). INVERSE HYPERBOLIC FUNCTION

   • Derivatives
   • Integral

10. (k). TECHNIQUE OF INTEGRATION

    • Completing the Square
      • When to use?
        • no substitution available
        • a quadratic function at denominator
    • Trigonometric Identities
      • When to use?
        • summation of two terms power two
        • different angle
    • Improper Fractions
      • When to use?
        • you have improper fraction
    • Separating Fractions
      • When to use?
        • if the fracion can be separated and will result to a simpler integrand
    • Multiplying by a Form of 1
      • used to multiply expression by some term divided by itself to simplify expression to a expression that can be integrated
    • Eliminating Square Roots
      • When to use?
        • a trigonometric function in the square root
        • trigonometric simplified to squared trigonometric form so that square root can be eliminated

11. (m). INTEGRATION BY PARTS

    • How to use?
      • 1. separate two functions into u and v.
      • 2. Integrate both sides with respect to x.
      • 3. Rearrange the term
    • Tabular Integration
      • u can be differentiated repeatedly until become ZERO

12. (n).RATIONAL FUNCTIONS BY PARTIAL FRACTIONS

    • Rules of Partial Frctions
      • numerator must be of lower degree than denominator. If not, divide out long division
      • factors the denominator into its prime actors to determine the shapre of the partial fractions
      • linear factor (ax+b) gives a partial fraction form A/(ax+b)
      • factors (ax+b)^2 gives a 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
      • quadratic factor (ax^2 + bx + c) give partial fraction of the form (Ax+B)/(ax^2 + bx + c)

13. (o). TRIGONOMETRIC INTEGRALS

    • To integrate ∫sin^2 x and ∫cos^2 x dx
      • express function in terms of the cosine of the half angle
    • To integrate ∫sin^3 x and ∫cos^3 x dx
      • release one factor one of the factors sin x and convert the remaining sin^2 x into (1-cos^2 x)
    • To integrate ∫sin^4 x and ∫cos^4 x dx
      • separate the function into sin^2 x and sin^2 x and then convert into (1- cos^2 x)
    • To integrate ∫ sin^5 and ∫ cos^5 x dx
      • separate the function into (sin^2 x)^2 and sin x
    • To integrate ∫ sin^n x and ∫ cos^n x dx
      • Use Reduction Formula
        • For ∫sin^n dx
        • For ∫cos ^n dx
    • Products of Powers of Sines and Cosines
      • ∫sin^m x cos^n x dx
        • CASE 1
          • if m is odd, m is 2k + 1 and use sin^2= 1-cos^2 x. Then combine the single sin x with dx and set sin x dx equal to -d(cos x)
        • CASE 2
          • if m is even and n is odd, n as 2k + 1 and use cos^2 x = 1 - sin^2 x . Then combine the single cos x with dx and set cos x equal to d(sin x)
        • CASE 3
          • if both m and n are even, use half angle identities to reduce the integrand to one in lower powers of cos 2x
    • Product of Sines and Cosines
    • Integrals of the form ∫(1/(a+b sin^2 x + c cos^2 x)) dx
      • substitute t= tan x
    • Integrals of the form ∫ (1/(a+b sin x + c cos x)) dx
      • substitute t= tan x/2

14. (p). TRIGONOMETRIC SUBSTITION

    • Replace the variable of intehratino by a trigonometric function

15. (q). IMPROPER INTEGRAL

    • Infinite Limits of Integration Type I
      • Upper Limit
      • Both Limits
      • Lower Limit
    • Infinite Limits of Integration Type II
      • Lower Endpoint
      • Upper Endpoint
      • Interior point

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