Improper Integrals (Cambridge International AS Mathematics)

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:

  • $\int_{0}^{5} \frac{1}{ \sqrt[3]{x} } d x$
  • $integral_{1}^{infinity} \frac{1}{x} ^{3} d x$

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

Exam Hint

• 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.