# CIE A Level Maths: Pure 3复习笔记5.1.4 Substitution (Reverse Chain Rule)

### Substitution (Reverse Chain Rule)

#### What is integration by substitution?

• Make sure you are familiar with Chain Rule and Reverse Chain Rule
• In more awkward problems it is harder to spot the reverse chain rule
• It is possible to use a substitution
• These kinds of substitutions usually will not be given
• The substitution is deemed ‘obvious’

#### How do I integrate when a substitution is not given?

• Look to substitute the ‘second’ (rather than the ‘main’) function
• STEP 1: Determine the substitution
• What is the ‘main’ function? ‘second’ function?
• STEP 2: Differentiate the substitution and rearrange
• du/dx here can be treated like a fraction (eg × dx to get rid of fractions)
• STEP 3: Replace all parts of the integral
• all x terms should be replaced with equivalent u terms, including dx
• if a definite integral change the limits from x to u too
• STEP 4: Integrate and either …
• …substitute x back inor
• … evaluate the definite integral using the u limits
• STEP 5: Find c, the constant of integration, if needed

#### Exam Tip

• A lot of the “work” in these problems happens in your head :
• what is the ‘main’ function?
• what should the substitution be?
• are there any numbers to ‘adjust’ and ‘compensate’ for?
• Be sure that what you write down is clear and easy to follow, and remember that you can check your final answer by differentiating it.