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IB DP Maths: AA SL复习笔记5.4.2 Techniques of Integration

Category: IB课程, 教材笔记, 福利干货 Date: 2022年7月18日 下午5:27

Integrating Composite Functions (ax+b)

What is a composite function?

  • A?composite?function?involves one function being applied after another
  • A composite function may be described as a “function of a function”

How do I integrate linear (ax+b)?functions?

Exam Tip

  • Although the specific formulae in this revision note aren't given to you in the formula booklet, they can all be derived from the reverse chain rule (see below)
  • The formula booklet contains almost all of the information you will need to apply to reverse chain rule and derive these in the exam, make sure you have it open on the right pages whilst you are working

Worked Example

5-4-2-ib-sl-aa-only-we1-soltn-a

5-4-2-ib-sl-aa-only-we1-soltn-b

Reverse Chain Rule

What is reverse chain rule?

  • The?Chain Rule?is a way of differentiating two (or more) functions
  • Reverse?Chain?Rule?(RCR) refers to?integrating?by?inspection
    • spotting that chain rule would be used in the reverse (differentiating) process

How do I know when to use reverse chain rule?

  • Reverse?chain?rule?is used when we have the?product?of a?composite?function?and the?derivative?of its?second?function
  • Integration is trickier than differentiation; many of the shortcuts do not work

How do I integrate using reverse chain rule?

  • If the product?can?be identified, the?integration?can be done “by?inspection”
    • there may be some “adjusting?and?compensating” to do
  • Notice a lot of the "adjust?and?compensate?method” happens mentally
    • this is indicated in the steps below by quote marks

Exam Tip

  • Before the exam, practice this until you are confident with the pattern and do not need to worry about the formula or steps anymore
    • This will save time in the exam
  • You can always check your work by differentiating, if you have time

Worked Example

iiq~htJ9_5-4-2-ib-sl-aa-only-we2-soltn

 

Substitution: Reverse Chain Rule

What is integration by substitution?

  • When reverse chain rule is difficult to spot or awkward to use then?integration?by?substitution?can be used

How do I integrate using substitution?

Exam Tip

  • Use your GDC to check the value of a definite integral, even in cases where working needs to be shown

Worked Example5-4-2-ib-sl-aa-only-we3-soltn-a

5-4-2-ib-sl-aa-only-we3-soltn-b

 

 

 

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