格式练习：定积分的计算过程(?)

原文链接

计算\displaystyle \int ^{2}_{-2} -\dfrac{1}{2}x^2+\dfrac{8}{3}\, \mathrm{d}x.

解：

先计算其不定积分。

\displaystyle \int -\dfrac{1}{2}x^2+\dfrac{8}{3}\, \mathrm{d}x

=-\displaystyle\int \dfrac{1}{2}x^2\,\mathrm{d}x+\displaystyle\int \dfrac{8}{3}\,\mathrm{d}x

=-\dfrac{x^3}{6}+\dfrac{8}{3}x

则

\displaystyle \int ^{2}_{-2} -\dfrac{1}{2}x^2+\dfrac{8}{3}\, \mathrm{d}x

=-\dfrac{x^3}{6}+\dfrac{8}{3}x \,\bigg|^{2}_{-2}

=-\dfrac{2^3}{6}+\dfrac{8}{3}\times 2-\left(-\dfrac{\left(-2\right)^3}{6}+\dfrac{8}{3}\times\left(-2\right)\right)

=8

Lake桑

2019.2.3


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