Integral of $$$\frac{x^{2}}{x^{6} + 2}$$$

Find $$$\int \frac{x^{2}}{x^{6} + 2}\, dx$$$.

Let $$$u=x^{3}$$$.

Then $$$du=\left(x^{3}\right)^{\prime }dx = 3 x^{2} dx$$$ (steps can be seen »), and we have that $$$x^{2} dx = \frac{du}{3}$$$.

The integral becomes

$${\color{red}{\int{\frac{x^{2}}{x^{6} + 2} d x}}} = {\color{red}{\int{\frac{1}{3 \left(u^{2} + 2\right)} d u}}}$$

Apply the constant multiple rule $$$\int c f{\left(u \right)}\, du = c \int f{\left(u \right)}\, du$$$ with $$$c=\frac{1}{3}$$$ and $$$f{\left(u \right)} = \frac{1}{u^{2} + 2}$$$:

$${\color{red}{\int{\frac{1}{3 \left(u^{2} + 2\right)} d u}}} = {\color{red}{\left(\frac{\int{\frac{1}{u^{2} + 2} d u}}{3}\right)}}$$

Let $$$v=\frac{\sqrt{2} u}{2}$$$.

Then $$$dv=\left(\frac{\sqrt{2} u}{2}\right)^{\prime }du = \frac{\sqrt{2}}{2} du$$$ (steps can be seen »), and we have that $$$du = \sqrt{2} dv$$$.

The integral can be rewritten as

$$\frac{{\color{red}{\int{\frac{1}{u^{2} + 2} d u}}}}{3} = \frac{{\color{red}{\int{\frac{\sqrt{2}}{2 \left(v^{2} + 1\right)} d v}}}}{3}$$

Apply the constant multiple rule $$$\int c f{\left(v \right)}\, dv = c \int f{\left(v \right)}\, dv$$$ with $$$c=\frac{\sqrt{2}}{2}$$$ and $$$f{\left(v \right)} = \frac{1}{v^{2} + 1}$$$:

$$\frac{{\color{red}{\int{\frac{\sqrt{2}}{2 \left(v^{2} + 1\right)} d v}}}}{3} = \frac{{\color{red}{\left(\frac{\sqrt{2} \int{\frac{1}{v^{2} + 1} d v}}{2}\right)}}}{3}$$

The integral of $$$\frac{1}{v^{2} + 1}$$$ is $$$\int{\frac{1}{v^{2} + 1} d v} = \operatorname{atan}{\left(v \right)}$$$:

$$\frac{\sqrt{2} {\color{red}{\int{\frac{1}{v^{2} + 1} d v}}}}{6} = \frac{\sqrt{2} {\color{red}{\operatorname{atan}{\left(v \right)}}}}{6}$$

Recall that $$$v=\frac{\sqrt{2} u}{2}$$$:

$$\frac{\sqrt{2} \operatorname{atan}{\left({\color{red}{v}} \right)}}{6} = \frac{\sqrt{2} \operatorname{atan}{\left({\color{red}{\left(\frac{\sqrt{2} u}{2}\right)}} \right)}}{6}$$

Recall that $$$u=x^{3}$$$:

$$\frac{\sqrt{2} \operatorname{atan}{\left(\frac{\sqrt{2} {\color{red}{u}}}{2} \right)}}{6} = \frac{\sqrt{2} \operatorname{atan}{\left(\frac{\sqrt{2} {\color{red}{x^{3}}}}{2} \right)}}{6}$$

Therefore,

$$\int{\frac{x^{2}}{x^{6} + 2} d x} = \frac{\sqrt{2} \operatorname{atan}{\left(\frac{\sqrt{2} x^{3}}{2} \right)}}{6}$$

Add the constant of integration:

$$\int{\frac{x^{2}}{x^{6} + 2} d x} = \frac{\sqrt{2} \operatorname{atan}{\left(\frac{\sqrt{2} x^{3}}{2} \right)}}{6}+C$$

$$$\int \frac{x^{2}}{x^{6} + 2}\, dx = \frac{\sqrt{2} \operatorname{atan}{\left(\frac{\sqrt{2} x^{3}}{2} \right)}}{6} + C$$$A