题目

Problem

Find

\begin{align*} \int\left(12x^3+\frac{1}{6\sqrt{x}}-\frac{3}{2x^4}\right)\,dx \end{align*}

giving each term in simplest form.

(5)

解答

展开

先把根号和分式写成指数形式，再逐项积分。每一项都用规则

\begin{align*} \int x^n\,dx=\frac{x^{n+1}}{n+1}+c \end{align*}

其中 $n\ne -1$ 。

展开

\begin{align*} \int\left(12x^3+\frac{1}{6\sqrt{x}}-\frac{3}{2x^4}\right)\,dx =&\,\int\left(12x^3+\frac16x^{-1/2}-\frac32x^{-4}\right)\,dx\\ =&\,12\cdot\frac{x^4}{4} +\frac16\cdot\frac{x^{1/2}}{1/2} -\frac32\cdot\frac{x^{-3}}{-3} +c\\ =&\,3x^4+\frac13x^{1/2}+\frac12x^{-3}+c\\ =&\,3x^4+\frac13\sqrt{x}+\frac{1}{2x^3}+c. \end{align*}