IB DP Maths: AI HL复习笔记5.4.2 Techniques of Integration

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?

• All the above can be deduced using reverse chain rule
  • However, spotting them can make solutions more efficient

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 workHow 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

• Differentiation can be used as a means of checking the final answer
• After some practice, you may find Step 2 is not needed
  • Do use it on more awkward questions (negatives and fractions!)
• If the product cannot easily be identified, use substitution

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?