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eMathHelp 数学求解器 – 免费分步计算器
分步求解数学问题
您的输入
求$$$\int \frac{3}{x^{2} + 2}\, dx$$$。
解答
对 $$$c=3$$$ 和 $$$f{\left(x \right)} = \frac{1}{x^{2} + 2}$$$ 应用常数倍法则 $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$：
$${\color{red}{\int{\frac{3}{x^{2} + 2} d x}}} = {\color{red}{\left(3 \int{\frac{1}{x^{2} + 2} d x}\right)}}$$
设$$$u=\frac{\sqrt{2}}{2} x$$$。
则$$$du=\left(\frac{\sqrt{2}}{2} x\right)^{\prime }dx = \frac{\sqrt{2}}{2} dx$$$ (步骤见»)，并有$$$dx = \sqrt{2} du$$$。
因此，
$$3 {\color{red}{\int{\frac{1}{x^{2} + 2} d x}}} = 3 {\color{red}{\int{\frac{\sqrt{2}}{2 \left(u^{2} + 1\right)} d u}}}$$
对 $$$c=\frac{\sqrt{2}}{2}$$$ 和 $$$f{\left(u \right)} = \frac{1}{u^{2} + 1}$$$ 应用常数倍法则 $$$\int c f{\left(u \right)}\, du = c \int f{\left(u \right)}\, du$$$：
$$3 {\color{red}{\int{\frac{\sqrt{2}}{2 \left(u^{2} + 1\right)} d u}}} = 3 {\color{red}{\left(\frac{\sqrt{2} \int{\frac{1}{u^{2} + 1} d u}}{2}\right)}}$$
$$$\frac{1}{u^{2} + 1}$$$ 的积分为 $$$\int{\frac{1}{u^{2} + 1} d u} = \operatorname{atan}{\left(u \right)}$$$:
$$\frac{3 \sqrt{2} {\color{red}{\int{\frac{1}{u^{2} + 1} d u}}}}{2} = \frac{3 \sqrt{2} {\color{red}{\operatorname{atan}{\left(u \right)}}}}{2}$$
回忆一下 $$$u=\frac{\sqrt{2}}{2} x$$$:
$$\frac{3 \sqrt{2} \operatorname{atan}{\left({\color{red}{u}} \right)}}{2} = \frac{3 \sqrt{2} \operatorname{atan}{\left({\color{red}{\frac{\sqrt{2}}{2} x}} \right)}}{2}$$
因此，
$$\int{\frac{3}{x^{2} + 2} d x} = \frac{3 \sqrt{2} \operatorname{atan}{\left(\frac{\sqrt{2} x}{2} \right)}}{2}$$
加上积分常数：
$$\int{\frac{3}{x^{2} + 2} d x} = \frac{3 \sqrt{2} \operatorname{atan}{\left(\frac{\sqrt{2} x}{2} \right)}}{2}+C$$
答案
$$$\int \frac{3}{x^{2} + 2}\, dx = \frac{3 \sqrt{2} \operatorname{atan}{\left(\frac{\sqrt{2} x}{2} \right)}}{2} + C$$$A