Edexcel A Level Further Maths: Core Pure:復習筆記5.2.1 Improper Integrals
Improper Integrals
What are improper integrals?
- An?improper integral?is a definite integral where one or both of the limits is either:
- Positive or minus infinity
- A point where the function is undefined
- Consider the graph of
- It is undefined at the point?x = 0
- The integral of ?with a?limit of zero?would be an improper integral
- Consider the graph of
- Examples include:
How do we find the value of an improper integral??
- Use algebra to?replace the limit?which cannot be found with a variable
- E.g. let the undefined limit of zero be?a?or the infinite limit be?b
- Evaluate the integral?and substitute your chosen variable into the expression
- Consider what will happen to your answer as the value of your chosen variable?tends towards the limit
- E.g. what happens as?a?gets closer to zero or as?b?gets closer to infinity?
- Your final answer will be the value you get if you substitute this into your answer
- E.g. as?a?tends to zero?a2?tends to zero and so this part of your solution will be zero
- It is useful to remember as?a tends to infinity then
?tends to 0
Exam Tip
- Be careful if a limit of your integral is zero, always check to see if the function is defined at zero and if not treat it as an improper integral.
- Infinite limits will always be treated as improper integrals.
Worked Example
Find the following improper integrals, give your answers as exact values,
a)
b)
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