Improper Integrals (Edexcel A Level Further Maths: Core Pure): 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 $y=\frac{1}{\sqrt{x}}$

    • 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:

  • ${\int }_0^5\frac{1}{\sqrt[3]{x}}dx=\underset{a\to 0}{\lim }{\int }_a^5\frac{1}{\sqrt[3]{x}}dx$

  • ${\int }_1^{\infty }\frac{1}{x^3}dx=\underset{b\to \infty }{\lim }{\int }_1^b\frac{1}{x^3}dx$

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