$$$\frac{x^{2}}{x^{6} + 2}$$$ 的积分
您的输入
求$$$\int \frac{x^{2}}{x^{6} + 2}\, dx$$$。
解答
设$$$u=x^{3}$$$。
则$$$du=\left(x^{3}\right)^{\prime }dx = 3 x^{2} dx$$$ (步骤见»)，并有$$$x^{2} dx = \frac{du}{3}$$$。
积分变为
$${\color{red}{\int{\frac{x^{2}}{x^{6} + 2} d x}}} = {\color{red}{\int{\frac{1}{3 \left(u^{2} + 2\right)} d u}}}$$
对 $$$c=\frac{1}{3}$$$ 和 $$$f{\left(u \right)} = \frac{1}{u^{2} + 2}$$$ 应用常数倍法则 $$$\int c f{\left(u \right)}\, du = c \int f{\left(u \right)}\, du$$$：
$${\color{red}{\int{\frac{1}{3 \left(u^{2} + 2\right)} d u}}} = {\color{red}{\left(\frac{\int{\frac{1}{u^{2} + 2} d u}}{3}\right)}}$$
设$$$v=\frac{\sqrt{2} u}{2}$$$。
则$$$dv=\left(\frac{\sqrt{2} u}{2}\right)^{\prime }du = \frac{\sqrt{2}}{2} du$$$ (步骤见»)，并有$$$du = \sqrt{2} dv$$$。
因此，
$$\frac{{\color{red}{\int{\frac{1}{u^{2} + 2} d u}}}}{3} = \frac{{\color{red}{\int{\frac{\sqrt{2}}{2 \left(v^{2} + 1\right)} d v}}}}{3}$$
对 $$$c=\frac{\sqrt{2}}{2}$$$ 和 $$$f{\left(v \right)} = \frac{1}{v^{2} + 1}$$$ 应用常数倍法则 $$$\int c f{\left(v \right)}\, dv = c \int f{\left(v \right)}\, dv$$$：
$$\frac{{\color{red}{\int{\frac{\sqrt{2}}{2 \left(v^{2} + 1\right)} d v}}}}{3} = \frac{{\color{red}{\left(\frac{\sqrt{2} \int{\frac{1}{v^{2} + 1} d v}}{2}\right)}}}{3}$$
$$$\frac{1}{v^{2} + 1}$$$ 的积分为 $$$\int{\frac{1}{v^{2} + 1} d v} = \operatorname{atan}{\left(v \right)}$$$:
$$\frac{\sqrt{2} {\color{red}{\int{\frac{1}{v^{2} + 1} d v}}}}{6} = \frac{\sqrt{2} {\color{red}{\operatorname{atan}{\left(v \right)}}}}{6}$$
回忆一下 $$$v=\frac{\sqrt{2} u}{2}$$$:
$$\frac{\sqrt{2} \operatorname{atan}{\left({\color{red}{v}} \right)}}{6} = \frac{\sqrt{2} \operatorname{atan}{\left({\color{red}{\left(\frac{\sqrt{2} u}{2}\right)}} \right)}}{6}$$
回忆一下 $$$u=x^{3}$$$:
$$\frac{\sqrt{2} \operatorname{atan}{\left(\frac{\sqrt{2} {\color{red}{u}}}{2} \right)}}{6} = \frac{\sqrt{2} \operatorname{atan}{\left(\frac{\sqrt{2} {\color{red}{x^{3}}}}{2} \right)}}{6}$$
因此，
$$\int{\frac{x^{2}}{x^{6} + 2} d x} = \frac{\sqrt{2} \operatorname{atan}{\left(\frac{\sqrt{2} x^{3}}{2} \right)}}{6}$$
加上积分常数：
$$\int{\frac{x^{2}}{x^{6} + 2} d x} = \frac{\sqrt{2} \operatorname{atan}{\left(\frac{\sqrt{2} x^{3}}{2} \right)}}{6}+C$$
答案
$$$\int \frac{x^{2}}{x^{6} + 2}\, dx = \frac{\sqrt{2} \operatorname{atan}{\left(\frac{\sqrt{2} x^{3}}{2} \right)}}{6} + C$$$A