题目

Problem

Find

\begin{align*} \int \frac{4x^5+3}{2x^2}\,dx \end{align*}

giving your answer in simplest form.

(5)

解答

展开

分母是单项式，所以先把分子每一项分别除以 $2x^2$ 。这样可以把分式改写成幂函数之和，再逐项积分。

展开

First rewrite the integrand:

\begin{align*} \frac{4x^5+3}{2x^2} =&\,\frac{4x^5}{2x^2}+\frac3{2x^2}\\ =&\,2x^3+\frac32x^{-2}. \end{align*}

Therefore

\begin{align*} \int \frac{4x^5+3}{2x^2}\,dx =&\,\int \left(2x^3+\frac32x^{-2}\right)\,dx\\ =&\,\frac12x^4-\frac32x^{-1}+c\\ =&\,\frac12x^4-\frac3{2x}+c. \end{align*}