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CIE 9231 2024 June Paper 23 Q1

A Level / CIE / FP2

CIE 9231 2024 June Paper 23 Paper · Question 1

题目

Problem

Find the exact value of

∫27/214x−x2−1 dx.\int_2^{7/2}\frac{1}{\sqrt{4x-x^2-1}}\,\mathrm{d}x.
[5]
题目中文翻译

求

∫27/214x−x2−1 dx\int_2^{7/2}\frac{1}{\sqrt{4x-x^2-1}}\,\mathrm{d}x

的精确值。

(5)
原卷通用作答要求

Answer all questions. You should use a calculator where appropriate.

You must show all necessary working clearly; no marks will be given for unsupported answers from a calculator.

Give non-exact numerical answers correct to 3 significant figures, or 1 decimal place for angles in degrees, unless a different level of accuracy is specified in the question.

请回答所有题目。适当时应使用计算器。必须清楚展示所有必要的计算过程；仅给出计算器答案而没有过程，不予评分。除非题目另有精度要求，非精确数值答案保留 3 位有效数字，以度为单位的角度保留 1 位小数。

解答

解法一

思路

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先把根号内配方为 3−(x−2)23-(x-2)^2，与反正弦的标准积分公式对应。上下限代入后，利用特殊角求精确值。

答题过程

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Completing the square gives

4x−x2−1=3−(x−2)2.4x-x^2-1=3-(x-2)^2.

For 2≤x≤7/22\le x\le7/2, we have 0≤x−2≤3/2<30\le x-2\le3/2<\sqrt3, so the square root is positive. Using the standard inverse sine integral,

∫27/214x−x2−1 dx= [sin⁡−1x−23]27/2= sin⁡−132−sin⁡−10= π3.\begin{align*} \int_2^{7/2}\frac{1}{\sqrt{4x-x^2-1}}\,\mathrm{d}x =&\,\bigg[\sin^{-1}\frac{x-2}{\sqrt3}\bigg]_2^{7/2}\\ =&\,\sin^{-1}\frac{\sqrt3}{2}-\sin^{-1}0\\ =&\,\boxed{\frac{\pi}{3}}. \end{align*}