Further Maths - Improper Integrals
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Flashcards
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.
What do you call an improper integral that does not exist?
Divergent.
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\[\int^\infty _ 0 e^{-x} dx\]What would you write instead to see if it has a defined value??
\[\lim_{t \to \infty} \int^t_0 e^{-x}dx\]2022-05-12
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\[\int^{5} _ {0} \frac{2}{(2-x)^\frac{1}{3}} \text{d}x\]How would you write this with limits as it is an improper integral??
\[\lim_{\alpha \to 2^{-}} \int^{\alpha}_{0} \frac{2}{(2-x)^\frac{1}{3}} \text{d} x + \lim_{\beta \to 2^{+}} \int^{5}_{\beta} \frac{2}{(2-x)^\frac{1}{3}} \text{d} x\]