题目 Problem Find ∫(8x35−23x4−1) dx\begin{align*} \int\left(\frac{8x^3}{5}-\frac{2}{3x^4}-1\right)\,dx \end{align*}∫(58x3−3x42−1)dx giving each term in your answer in its simplest form. (4) 解答 解法一 思路 展开 先把分式项写成负指数： 23x4=23x−4.\begin{align*} \frac{2}{3x^4}=\frac23x^{-4}. \end{align*}3x42=32x−4. 然后逐项积分。最后不要忘记积分常数。 答题过程 展开 ∫(8x35−23x4−1) dx= ∫(85x3−23x−4−1) dx= 85⋅x44−23⋅x−3−3−x+c= 25x4+29x−3−x+c= 25x4+29x3−x+c.\begin{align*} \int\left(\frac{8x^3}{5}-\frac{2}{3x^4}-1\right)\,dx =&\,\int\left(\frac85x^3-\frac23x^{-4}-1\right)\,dx\\ =&\,\frac85\cdot\frac{x^4}{4} -\frac23\cdot\frac{x^{-3}}{-3} -x+c\\ =&\,\frac25x^4+\frac29x^{-3}-x+c\\ =&\,\frac25x^4+\frac{2}{9x^3}-x+c. \end{align*}∫(58x3−3x42−1)dx====∫(58x3−32x−4−1)dx58⋅4x4−32⋅−3x−3−x+c52x4+92x−3−x+c52x4+9x32−x+c.
