Integration Techniques

Integration Techniques

1. Double Substitution

   • Find variables x and u
     • Substitute x on the original equation
       • Apply Distributy property
         • Solve the remaining integrals separately
   • It is used when one of the product cannot be canceled through any normal rule.

2. Partial Fractions

   • Factor the denominator
     • For every factor you are going to have an unknown.
       • Multiply everything times all of the factors, in order to cancel de denominators
         • Give values to x in order to cancel most of the letters and stay with one
           • Isolate the unknown
             • Repeat the process until you find every unknown.
               • *Sometimes you will need a system of equations to find some of the unknowns.
   • Used when the integral is expressed as a Fraction with a denominator which has several factors

3. Integration by Parts

   • Follow the LATE rule
     • LogarithmicAlgebraicTrigonometricExponential
     • Which ever comes first on LATE will be the "u"
       • De rest of the equation is "dv"
         • Integrate to find v
           • Apply the Integration by parts Formula
             • uv ∫vdu
               • Solve the ∫vdu integral
       • Find "du"
   • Used when we hae different kind of functions on the problem

4. Tabula Method

   • In some cases we can use a table as a shortcut on integration by parts
     • Examples: ∫x^nSin(ax+b)dx , ∫x^nCos(ax+b)dx
       • In such cases, u=x^n, the rest will be dc
     • Signs will start + and alternate
     • Calculate the derivative until it is 0
       • The solution is obtained adding the products of the entrance of the diagonals.

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