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Further Calculus

Quick questions on Improper integrals explained: H2 Further Mathematics

5short Q&A pairs drawn directly from our worked dot-point answer. For full context and worked exam questions, read the parent dot-point page.

What is two kinds of improper integral?
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An integral is improper if either:
What is evaluating by a limit?
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Replace the troublesome limit with a variable and take a limit:
What is q1?
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Explain why ∫1∞1x dx\displaystyle\int_1^{\infty}\dfrac{1}{x}\,dx diverges. [2 marks]
What is q2?
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State the condition on pp for ∫1∞1xp dx\displaystyle\int_1^{\infty}\dfrac{1}{x^p}\,dx to converge. [1 mark]
What is q3?
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Why is ∫011x dx\displaystyle\int_0^{1}\dfrac{1}{\sqrt{x}}\,dx improper? [1 mark]

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