國際數學圖譜

IAL 2021 June Q4

A Level / Edexcel / P1

IAL 2021 June Paper · Question 4

题目

Problem

Find

\begin{align*} \int \frac{(3\sqrt{x}+2)(x-5)}{4\sqrt{x}}\,dx \end{align*}

writing each term in simplest form.

(6)

解答

解法一

思路

展开

先展开分子，再逐项除以 $4\sqrt{x}$ 。把被积函数拆成幂函数后再积分。

答题过程

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First expand the numerator:

\begin{align*} (3\sqrt{x}+2)(x-5) =&\,3x\sqrt{x}-15\sqrt{x}+2x-10. \end{align*}

Therefore

\begin{align*} \frac{(3\sqrt{x}+2)(x-5)}{4\sqrt{x}} =&\,\frac{3x\sqrt{x}}{4\sqrt{x}} -\frac{15\sqrt{x}}{4\sqrt{x}} +\frac{2x}{4\sqrt{x}} -\frac{10}{4\sqrt{x}}\\ =&\,\frac34x-\frac{15}{4} +\frac12x^{1/2} -\frac52x^{-1/2}. \end{align*}

Now integrate:

\begin{align*} \int \frac{(3\sqrt{x}+2)(x-5)}{4\sqrt{x}}\,dx =&\,\int\left(\frac34x-\frac{15}{4} +\frac12x^{1/2} -\frac52x^{-1/2}\right)\,dx\\ =&\,\frac38x^2-\frac{15}{4}x +\frac12\cdot\frac{x^{3/2}}{3/2} -\frac52\cdot\frac{x^{1/2}}{1/2} +c\\ =&\,\frac38x^2-\frac{15}{4}x +\frac13x^{3/2} -5x^{1/2}+c. \end{align*}