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Substitution (Reverse Chain Rule) (CIE A Level Maths: Pure 3)
Revision Note
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’
- 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 in
or
- … evaluate the definite integral using the u limits
- …substitute x back in
- STEP 5: Find c, the constant of integration, if needed
Examiner 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.
Worked example
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