$$$\frac{x^{4}}{\sqrt{1 - x^{4}}}$$$ 的积分
您的输入
求$$$\int \frac{x^{4}}{\sqrt{1 - x^{4}}}\, dx$$$。
解答
该积分没有闭式表达式：
$${\color{red}{\int{\frac{x^{4}}{\sqrt{1 - x^{4}}} d x}}} = {\color{red}{\left(\frac{x^{5} {{}_{2}F_{1}\left(\begin{matrix} \frac{1}{2}, \frac{5}{4} \\ \frac{9}{4} \end{matrix}\middle| {x^{4}} \right)}}{5}\right)}}$$
因此，
$$\int{\frac{x^{4}}{\sqrt{1 - x^{4}}} d x} = \frac{x^{5} {{}_{2}F_{1}\left(\begin{matrix} \frac{1}{2}, \frac{5}{4} \\ \frac{9}{4} \end{matrix}\middle| {x^{4}} \right)}}{5}$$
加上积分常数：
$$\int{\frac{x^{4}}{\sqrt{1 - x^{4}}} d x} = \frac{x^{5} {{}_{2}F_{1}\left(\begin{matrix} \frac{1}{2}, \frac{5}{4} \\ \frac{9}{4} \end{matrix}\middle| {x^{4}} \right)}}{5}+C$$
答案
$$$\int \frac{x^{4}}{\sqrt{1 - x^{4}}}\, dx = \frac{x^{5} {{}_{2}F_{1}\left(\begin{matrix} \frac{1}{2}, \frac{5}{4} \\ \frac{9}{4} \end{matrix}\middle| {x^{4}} \right)}}{5} + C$$$A