Further Maths - Improper Integrals

Created: April 27, 2021

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See Also

• [[Maths - Integration]]^A
• [[Maths - Integration by Substitution]]^A

Flashcards

What two things mean an integral is improper?

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• 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)?

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Convergent.

What do you call an improper integral that does not exist?

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Divergent.

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$${\int }_0^{\mathrm{\infty }}{e}^{-x}dx$$

What would you write instead to see if it has a defined value??

$$\underset{t\to \mathrm{\infty }}{lim}{\int }_0^t{e}^{-x}dx$$

2022-05-12

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$${\int }_0^5\frac{2}{(2-x{)}^{\frac{1}{3}}}\text{d}x$$

How would you write this with limits as it is an improper integral??

$$\underset{\alpha \to 2^{-}}{lim}{\int }_0^{\alpha }\frac{2}{(2-x{)}^{\frac{1}{3}}}\text{d}x+\underset{\beta \to 2^{+}}{lim}{\int }_{\beta }^5\frac{2}{(2-x{)}^{\frac{1}{3}}}\text{d}x$$

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Related posts

• [[Maths - Integration]]^A

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• [[Maths - Integration by Substitution]]^A

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• [[Further Maths - Syllabus]]^A

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• [[Further Maths - Integrating and Differentiating Inverse Trig Functions]]^A

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• [[MAT - Paper 2011 - Q1G]]^N

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• [[Course - Analysis III TT23]]^U

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