题目 Problem Find ∫(23x3−12x3+5) dx\begin{align*} \int\left(\frac23x^3-\frac{1}{2x^3}+5\right)\,dx \end{align*}∫(32x3−2x31+5)dx simplifying your answer. (4) 解答 解法一 思路 展开 把分式写成负指数，再逐项积分。 答题过程 展开 ∫(23x3−12x3+5) dx= ∫(23x3−12x−3+5) dx= 23⋅x44−12⋅x−2−2+5x+c= 16x4+14x−2+5x+c= 16x4+14x2+5x+c.\begin{align*} \int\left(\frac23x^3-\frac{1}{2x^3}+5\right)\,dx =&\,\int\left(\frac23x^3-\frac12x^{-3}+5\right)\,dx\\ =&\,\frac23\cdot\frac{x^4}{4} -\frac12\cdot\frac{x^{-2}}{-2} +5x+c\\ =&\,\frac16x^4+\frac14x^{-2}+5x+c\\ =&\,\frac16x^4+\frac{1}{4x^2}+5x+c. \end{align*}∫(32x3−2x31+5)dx====∫(32x3−21x−3+5)dx32⋅4x4−21⋅−2x−2+5x+c61x4+41x−2+5x+c61x4+4x21+5x+c.
