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?
  • 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,