Improper Integrals (Edexcel A Level Further Maths): Revision Note

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