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IAL 2025 Jan Q1

A Level / Edexcel / P1

IAL 2025 Jan Paper · Question 1

题目

Problem

Find

∫(8x3−6x−25x3) dx,\begin{align*} \int\left(8x^3-6\sqrt{x}-\frac{2}{5x^3}\right)\,dx, \end{align*}

giving your answer in simplest form.

(4)

解答

解法一

思路

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先把根式和分式都写成指数形式：

x=x1/2,1x3=x−3.\begin{align*} \sqrt{x}=x^{1/2},\qquad \frac{1}{x^3}=x^{-3}. \end{align*}

然后逐项积分，最后不要忘记积分常数。

答题过程

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Write the integrand using powers of xx:

8x3−6x−25x3=8x3−6x1/2−25x−3.\begin{align*} 8x^3-6\sqrt{x}-\frac{2}{5x^3} =8x^3-6x^{1/2}-\frac25x^{-3}. \end{align*}

So

∫(8x3−6x−25x3) dx= ∫(8x3−6x1/2−25x−3) dx= 8⋅x44−6⋅x3/23/2−25⋅x−2−2+c= 2x4−4x3/2+15x−2+c= 2x4−4x3/2+15x2+c.\begin{align*} \int\left(8x^3-6\sqrt{x}-\frac{2}{5x^3}\right)\,dx =&\,\int\left(8x^3-6x^{1/2}-\frac25x^{-3}\right)\,dx\\ =&\,8\cdot\frac{x^4}{4} -6\cdot\frac{x^{3/2}}{3/2} -\frac25\cdot\frac{x^{-2}}{-2}+c\\ =&\,2x^4-4x^{3/2}+\frac15x^{-2}+c\\ =&\,2x^4-4x^{3/2}+\frac{1}{5x^2}+c. \end{align*}