Edexcel A Level Maths: Pure:复习笔记8.2.8 Integration by Parts
Integration by Parts
What is Integration by Parts?
- For integrating the product of two functions - reverse product rule
- Crucially the product is made from?u?and?dv/dx?(rather than?u?and?v)
- Alternative notation may be used
How do I use Integration by Parts?
- The hardest part is choosing?u?and?dv/dx?as there is no method for doing so
- u, ideally, becomes?simpler?when differentiated but this is not always possible
- dv/dx?should be a function that can be integrated fairly easily
- Be wary of functions that ‘cycle’/’repeat’ when differentiated/integrated
- ex?→?ex
- sin x?→?cos x?→?-sin x?→?-cos x?→?sin x
- STEP 1: Choose u and v’, find u’ and v
- STEP 2: Apply Integration by Parts
- Simplify anything straightforward
- STEP 3: Do the ‘second’ integral
- If an?indefinite?integral remember “+c”, the?constant?of?integration
- STEP 4: Simplify and/or apply limits
What happens if I cannot integrate?v × du/dx?
- It is possible integration by parts may need to be applied more than once
ln x
- A classic ‘set piece’ in almost every A level maths textbook ever written!
- In general, rewriting?f(x)?as?1×f(x)?can be a powerful problem-solving technique
- This could be a question in the exam
How do I find a definite integral using parts?
Exam Tip
- Always think about what an elegant, slick, professional maths solution looks like – solutions normally get more complicated at first but quickly get simpler.
- If your work is continuing to get more complicated, stop and check for an error.
- Try to develop a sense of ‘having gone too far down the wrong path’.
- This general advice is useful to remember:
-
- Is the second integral harder than the first?
- Try swapping your choice of?u?and?dv/dx
- It is rare to have to apply integration by parts more than twice
Worked Example
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