Techniques of Integration

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how to attempt certain questions
hamidymuhammad
Mind Map by hamidymuhammad, updated more than 1 year ago
hamidymuhammad
Created by hamidymuhammad about 8 years ago
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Resource summary

Techniques of Integration
  1. basic substitution method
    1. 1) identify the U
      1. 2) differentiate U and change the question to the form of U
        1. 3) integrate as usual
        2. completing the square,CSM (when quadratic function at denominator)
          1. 1) simplify denominator using CSM
            1. 2) integrate the equation
              1. *answer usually in the form of inverse trigo.
              2. trigonometric identities (when there is summation of 2 terms ^2 @ different angle)
                1. 1) expand the equation
                  1. 2) apply the trigonometric identitiesl
                    1. 3) make sure every term can be integrated
                      1. 4) integrate each term as usual.
                      2. improper fraction (when degree of numerator>/= degree of denominator)
                        1. 1) use long division method to simplify the equation
                          1. 2) integrate the simplified eq. as usual
                          2. separating fractions (when the fraction can be separated)
                            1. 1) separate the function into 2 eq. with same denominator.
                              1. 2) simplify then integrate each term
                              2. multiplying by a form of 1
                                1. 1) multiply by a form of 1 or its conjugate
                                  1. 2) simplify the expression
                                    1. 3) integrate the expression as usual.
                                    2. eliminating square roots(when trigo function is in the square root)
                                      1. 1) simplify to a squared trigonometric form
                                        1. 2) eliminate the square root
                                          1. 3) integrate as usual
                                          2. integration by parts
                                            1. in the form of
                                              1. 1) identify u and dv
                                                1. use ILATE
                                                2. 2) substitute into the form
                                                3. tabular integration
                                                  1. 1) diff u until becomes 0
                                                    1. 2) combine the product of fn
                                                  2. integration by partial fractions
                                                    1. non-repeated linear factors
                                                      1. 1) simplify the deniminator
                                                        1. 2) separate the denominator
                                                          1. 3) find A and B
                                                            1. use Heaviside "cover up" method
                                                          2. repeated linear factors
                                                            1. irreducible quadratic factors
                                                              1. improper fraction
                                                              2. trigonometric integrals
                                                                1. when the trigo fn is to the power of an even no.
                                                                  1. use half angle formula then integrate using basic subs. method
                                                                  2. when trigo fn is to the power of an odd no.
                                                                    1. 1) release one of the factor. convert the remaining using identities.
                                                                      1. 2) integrate each term.
                                                                      2. reduction formula
                                                                        1. to integrate sin^n x and cos^n x
                                                                        2. product of powers of sines and cosines.
                                                                          1. both odd
                                                                            1. one odd one even
                                                                              1. both even
                                                                              2. t-substitution (t=tan x)
                                                                                1. 1) change the eq. in terms of t
                                                                                  1. 2) integrate the fn
                                                                                    1. 3)convert back the t into tan x
                                                                                  2. trigonometric substitution
                                                                                    1. 1)convert the variable in terms of theta
                                                                                      1. 2) integrate as usual
                                                                                        1. 3) change the theta back into variable form
                                                                                        2. improper integral
                                                                                          1. 1) sketch the graph
                                                                                            1. 2) change into limit
                                                                                              1. 3) integrate the equation
                                                                                                1. By: Hamidy and Anas (set7)
                                                                                                2. 4) solve the limit.
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