Integral of $$$x \operatorname{acos}{\left(1 - x^{4} \right)}$$$

Find $$$\int x \operatorname{acos}{\left(1 - x^{4} \right)}\, dx$$$.

For the integral $$$\int{x \operatorname{acos}{\left(1 - x^{4} \right)} d x}$$$, use integration by parts $$$\int \operatorname{u} \operatorname{dv} = \operatorname{u}\operatorname{v} - \int \operatorname{v} \operatorname{du}$$$.

Let $$$\operatorname{u}=\operatorname{acos}{\left(1 - x^{4} \right)}$$$ and $$$\operatorname{dv}=x dx$$$.

Then $$$\operatorname{du}=\left(\operatorname{acos}{\left(1 - x^{4} \right)}\right)^{\prime }dx=\frac{4 x}{\sqrt{2 - x^{4}}} dx$$$ (steps can be seen ») and $$$\operatorname{v}=\int{x d x}=\frac{x^{2}}{2}$$$ (steps can be seen »).

Thus,

$${\color{red}{\int{x \operatorname{acos}{\left(1 - x^{4} \right)} d x}}}={\color{red}{\left(\operatorname{acos}{\left(1 - x^{4} \right)} \cdot \frac{x^{2}}{2}-\int{\frac{x^{2}}{2} \cdot \frac{4 x}{\sqrt{2 - x^{4}}} d x}\right)}}={\color{red}{\left(\frac{x^{2} \operatorname{acos}{\left(1 - x^{4} \right)}}{2} - \int{\frac{2 x^{3}}{\sqrt{2 - x^{4}}} d x}\right)}}$$

Let $$$u=2 - x^{4}$$$.

Then $$$du=\left(2 - x^{4}\right)^{\prime }dx = - 4 x^{3} dx$$$ (steps can be seen »), and we have that $$$x^{3} dx = - \frac{du}{4}$$$.

Therefore,

$$\frac{x^{2} \operatorname{acos}{\left(1 - x^{4} \right)}}{2} - {\color{red}{\int{\frac{2 x^{3}}{\sqrt{2 - x^{4}}} d x}}} = \frac{x^{2} \operatorname{acos}{\left(1 - x^{4} \right)}}{2} - {\color{red}{\int{\left(- \frac{1}{2 \sqrt{u}}\right)d u}}}$$

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

$$\frac{x^{2} \operatorname{acos}{\left(1 - x^{4} \right)}}{2} - {\color{red}{\int{\left(- \frac{1}{2 \sqrt{u}}\right)d u}}} = \frac{x^{2} \operatorname{acos}{\left(1 - x^{4} \right)}}{2} - {\color{red}{\left(- \frac{\int{\frac{1}{\sqrt{u}} d u}}{2}\right)}}$$

Apply the power rule $$$\int u^{n}\, du = \frac{u^{n + 1}}{n + 1}$$$ $$$\left(n \neq -1 \right)$$$ with $$$n=- \frac{1}{2}$$$:

$$\frac{x^{2} \operatorname{acos}{\left(1 - x^{4} \right)}}{2} + \frac{{\color{red}{\int{\frac{1}{\sqrt{u}} d u}}}}{2}=\frac{x^{2} \operatorname{acos}{\left(1 - x^{4} \right)}}{2} + \frac{{\color{red}{\int{u^{- \frac{1}{2}} d u}}}}{2}=\frac{x^{2} \operatorname{acos}{\left(1 - x^{4} \right)}}{2} + \frac{{\color{red}{\frac{u^{- \frac{1}{2} + 1}}{- \frac{1}{2} + 1}}}}{2}=\frac{x^{2} \operatorname{acos}{\left(1 - x^{4} \right)}}{2} + \frac{{\color{red}{\left(2 u^{\frac{1}{2}}\right)}}}{2}=\frac{x^{2} \operatorname{acos}{\left(1 - x^{4} \right)}}{2} + \frac{{\color{red}{\left(2 \sqrt{u}\right)}}}{2}$$

Recall that $$$u=2 - x^{4}$$$:

$$\frac{x^{2} \operatorname{acos}{\left(1 - x^{4} \right)}}{2} + \sqrt{{\color{red}{u}}} = \frac{x^{2} \operatorname{acos}{\left(1 - x^{4} \right)}}{2} + \sqrt{{\color{red}{\left(2 - x^{4}\right)}}}$$

Therefore,

$$\int{x \operatorname{acos}{\left(1 - x^{4} \right)} d x} = \frac{x^{2} \operatorname{acos}{\left(1 - x^{4} \right)}}{2} + \sqrt{2 - x^{4}}$$

Add the constant of integration:

$$\int{x \operatorname{acos}{\left(1 - x^{4} \right)} d x} = \frac{x^{2} \operatorname{acos}{\left(1 - x^{4} \right)}}{2} + \sqrt{2 - x^{4}}+C$$

$$$\int x \operatorname{acos}{\left(1 - x^{4} \right)}\, dx = \left(\frac{x^{2} \operatorname{acos}{\left(1 - x^{4} \right)}}{2} + \sqrt{2 - x^{4}}\right) + C$$$A