Integration Techniques

Integration Techniques

1. Double substitution

   • This method is one of the many methods for evaluation of integrals in calculus
   • ∫f(g(x))g'(x)dx = ∫f(u)du
   • Steps:
     • 1) Substitute: Begin by changing the integral from a function of "x" to a function of "u". To do this, you have to identify the function g(x) in the integral that you would replace with "u". Let "u" equal to g(x). Differentiate "u" with respect to "x" and solve for g'(x)dx in terms of du. Now you are ready to make a complete substitution in the original integral. Substitute "u" in place of g(x) in the given integral. Also, substitute the expression you obtained in terms of du for g'(x)dx. Your new integral should be completely in terms of "u" and not in terms of "x"
       • 2) Integrate: Evaluate the new integral with respect to "u"
         • 3) Replace: Replace "u" with g(x) in the integral solution

2. Partial fractions

   • This method is used when we are asked to take the integral of a rational expression of polynomials
   • Steps:
     • 1) Factor the bottom
       • 2) Write one partial fraction for each of those factors assigning variables in the numerator
         • 3) Multiply through by the bottom so we no longer have fractions
           • 4) Now find the variables, substituting "x" solving for "x" in the factors
             • 5) Integrate the fractions that are left

3. Integration by parts

   • This is a special method of integration that if often useful when two functions are multiplied together, but is also herlpful in other ways
   • ∫udv = uv - ∫vdu
   • Steps:
     • 1) Write the given integral
       • 2) Introduce the intermediary functions u(x) and v(x)
         • 3) Use the formula
           • 4) Integrate
   • In some cases you will need to use the tabulation method
     • Example: ∫[x^2(e^x)]dx
       • "U" is x^2, because of LATE
         • You gotta calculate the derivative until you obtain 0 as a derivative
           • So, you obtain as solution: x^2(e^x) - 2x(e^x) + C
   • L (logarithmic) A (algebraic) T (trigonometric E (exponential)

4. Edgar L. Ramírez Muñoz (A01283163)