Further Maths - Improper Integrals
Pearson Edexcel Further Mathematics 2022
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When an integral is improper
What two things mean an integral is improper?
- One or both of its limits are infinite
- It is undefined anywhere in $[a, b]$.
What do you call an improper integral that exists (has a defined value)?
Convergent.
Rewriting with a limit
\[\int^\infty _ 0 e^{-x} dx\]
What would you write instead to see if it has a defined value?
\[\lim _ {t \to \infty} \int^t _ 0 e^{-x}dx\]
\[\int^{5} _ {0} \frac{2}{(2-x)^\frac{1}{3}} \text{d}x\]
How would you write this with limits as it is an improper integral?
\[\lim _ {\alpha \to 2^{-}} \int^{\alpha} _ {0} \frac{2}{(2-x)^\frac{1}{3}} \text{d} x
+
\lim _ {\beta \to 2^{+}} \int^{5} _ {\beta} \frac{2}{(2-x)^\frac{1}{3}} \text{d} x\]