# Edexcel A Level Further Maths: Core Pure:复习笔记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
• 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?
• 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.

a)

b)