题目

Problem

Find

\begin{align*} \int\left(\frac12x^3+\frac{3}{\sqrt{x}}-4\right)\,dx \end{align*}

writing your answer in simplest form.

(4)

解答

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先把 $\dfrac{3}{\sqrt{x}}$ 写成指数形式 $3x^{-1/2}$ ，然后逐项积分。最后不要忘记积分常数。

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Write the integrand using powers of $x$ :

\begin{align*} \frac12x^3+\frac{3}{\sqrt{x}}-4 =\frac12x^3+3x^{-1/2}-4. \end{align*}

\begin{align*} \int\left(\frac12x^3+\frac{3}{\sqrt{x}}-4\right)\,dx =&\,\int\left(\frac12x^3+3x^{-1/2}-4\right)\,dx\\ =&\,\frac12\cdot\frac{x^4}{4} +3\cdot\frac{x^{1/2}}{1/2}-4x+c\\ =&\,\frac18x^4+6x^{1/2}-4x+c. \end{align*}