题目
Problem
Find the exact value of
∫ 1 5 2 1 x 2 − 2 x + 5 d x , \int_{1}^{\frac{5}{2}} \frac{1}{\sqrt{x^2 - 2x + 5}}\,\mathrm{d}x, ∫ 1 2 5 x 2 − 2 x + 5 1 d x , giving your answer in logarithmic form.
[6]
题目中文翻译
求
∫ 1 5 2 1 x 2 − 2 x + 5 d x \int_{1}^{\frac{5}{2}} \frac{1}{\sqrt{x^2 - 2x + 5}}\,\mathrm{d}x ∫ 1 2 5 x 2 − 2 x + 5 1 d x 的精确值，并用对数形式给出答案。
(6)
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，识别反双曲正弦的积分形式。代入端点后再改写为对数。
答题过程
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Since x 2 − 2 x + 5 = ( x − 1 ) 2 + 4 x^2-2x+5=(x-1)^2+4 x 2 − 2 x + 5 = ( x − 1 ) 2 + 4 u = ( x − 1 ) / 2 u=(x-1)/2 u = ( x − 1 ) /2 d x = 2 d u \mathrm{d}x=2\,\mathrm{d}u d x = 2 d u 0 0 0 3 / 4 3/4 3/4
∫ 1 5 / 2 1 ( x − 1 ) 2 + 4 d x = ∫ 0 3 / 4 1 1 + u 2 d u . \int_1^{5/2}\frac1{\sqrt{(x-1)^2+4}}\,\mathrm{d}x
=\int_0^{3/4}\frac1{\sqrt{1+u^2}}\,\mathrm{d}u. ∫ 1 5/2 ( x − 1 ) 2 + 4 1 d x = ∫ 0 3/4 1 + u 2 1 d u . Using the inverse hyperbolic sine integral,
∫ 1 5 / 2 1 x 2 − 2 x + 5 d x = [ arsinh ( x − 1 2 ) ] 1 5 / 2 = arsinh 3 4 = ln ( 3 4 + 1 + 9 16 ) = ln 2 . \begin{align*}
\int_1^{5/2}\frac1{\sqrt{x^2-2x+5}}\,\mathrm{d}x
=&\,\bigg[\operatorname{arsinh}\bigg(\frac{x-1}{2}\bigg)\bigg]_1^{5/2}\\
=&\,\operatorname{arsinh}\frac34\\[4mm]
=&\,\ln\bigg(\frac34+\sqrt{1+\frac9{16}}\bigg)\\[4mm]
=&\,\boxed{\ln2}.
\end{align*} ∫ 1 5/2 x 2 − 2 x + 5 1 d x = = = = [ arsinh ( 2 x − 1 ) ] 1 5/2 arsinh 4 3 ln ( 4 3 + 1 + 16 9 ) ln 2 . 