A LevelMaths: Pure 1CIERevision Notes6. Integration6.1 Integration6.1.5 Improper Integrals

Improper Integrals (CIE A Level Maths: Pure 1)

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Amber

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21 August 2023

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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 $y = \frac{1}{\sqrt{x}}$
  - It is undefined at the point x = 0
  - The integral of $y = \frac{1}{\sqrt{x}}$ with a limit of zero would be an improper integral
Examples include:
- $\int_{0}^{5} \frac{1}{\sqrt[3]{x}} d x$
- $\int_{1}^{\infty} \frac{1}{x^{3}} d x$

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 $\frac{1}{a}$ 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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Author: Amber

Expertise: Maths

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.