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:
- $\int_{0}^{5} \frac{1}{\sqrt[3]{x}} d x = \lim_{a \to 0} \int_{a}^{5} \frac{1}{\sqrt[3]{x}} d x$
- $\int_{1}^{\infty} \frac{1}{x^{3}} d x = \lim_{b \to \infty} \int_{1}^{b} \frac{1}{x^{3}} d x$

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

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

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.

Find the following improper integrals, give your answers as exact values,

$\int_{0}^{3} \frac{1}{2 \sqrt{x}} d x$

5-2-1-edx-a-fm-we1a-soltn

\int_{3}^{\infty} \frac{1}{2 x^{2}} d x

5-2-1-edx-a-fm-we1b-soltn

Improper Integrals (Edexcel A Level Further Maths: Core Pure)