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ASMaths: Pure 1CIERevision Notes6. Integration6.1 Integration6.1.5 Improper Integrals

Improper Integrals (CIE AS Maths: Pure 1)

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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 space y equals fraction numerator 1 over denominator square root of x end fraction
      • It is undefined at the point x = 0
      • The integral of  space y equals fraction numerator 1 over denominator square root of x end fraction with a limit of zero would be an improper integral
  • Examples include:
    • integral subscript 0 superscript 5 fraction numerator 1 over denominator cube root of x end fraction d x
    • integral subscript 1 superscript infinity 1 over x cubed d x

6-1-5-improper-integrals-diagram-1

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 1 over a tends to 0

Worked example

6-1-5-improper-integrals-we-solution-part-1

6-1-5-improper-integrals-we-solution-part-2

Examiner 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.

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    Amber

    Author: Amber

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    Amber gained a first class degree in Mathematics & Meteorology from the University of Reading before training to become a teacher. She is passionate about teaching, having spent 8 years teaching GCSE and A Level Mathematics both in the UK and internationally. Amber loves creating bright and informative resources to help students reach their potential.