$$$x^{4} \sqrt{1 - x^{4}}$$$ 的積分
您的輸入
求$$$\int x^{4} \sqrt{1 - x^{4}}\, dx$$$。
解答
此積分沒有閉式表示:
$${\color{red}{\int{x^{4} \sqrt{1 - x^{4}} d x}}} = {\color{red}{\left(\frac{x^{5} {{}_{2}F_{1}\left(\begin{matrix} - \frac{1}{2}, \frac{5}{4} \\ \frac{9}{4} \end{matrix}\middle| {x^{4}} \right)}}{5}\right)}}$$
因此，
$$\int{x^{4} \sqrt{1 - x^{4}} d x} = \frac{x^{5} {{}_{2}F_{1}\left(\begin{matrix} - \frac{1}{2}, \frac{5}{4} \\ \frac{9}{4} \end{matrix}\middle| {x^{4}} \right)}}{5}$$
加上積分常數：
$$\int{x^{4} \sqrt{1 - x^{4}} d x} = \frac{x^{5} {{}_{2}F_{1}\left(\begin{matrix} - \frac{1}{2}, \frac{5}{4} \\ \frac{9}{4} \end{matrix}\middle| {x^{4}} \right)}}{5}+C$$
答案
$$$\int x^{4} \sqrt{1 - x^{4}}\, dx = \frac{x^{5} {{}_{2}F_{1}\left(\begin{matrix} - \frac{1}{2}, \frac{5}{4} \\ \frac{9}{4} \end{matrix}\middle| {x^{4}} \right)}}{5} + C$$$A