$$$\frac{x^{21}}{x^{2} - 4}$$$ 的積分

求$$$\int \frac{x^{21}}{x^{2} - 4}\, dx$$$。

由於分子次數不小於分母次數，進行多項式長除法（步驟見»）:

$${\color{red}{\int{\frac{x^{21}}{x^{2} - 4} d x}}} = {\color{red}{\int{\left(x^{19} + 4 x^{17} + 16 x^{15} + 64 x^{13} + 256 x^{11} + 1024 x^{9} + 4096 x^{7} + 16384 x^{5} + 65536 x^{3} + 262144 x + \frac{1048576 x}{x^{2} - 4}\right)d x}}}$$

逐項積分：

$${\color{red}{\int{\left(x^{19} + 4 x^{17} + 16 x^{15} + 64 x^{13} + 256 x^{11} + 1024 x^{9} + 4096 x^{7} + 16384 x^{5} + 65536 x^{3} + 262144 x + \frac{1048576 x}{x^{2} - 4}\right)d x}}} = {\color{red}{\left(\int{262144 x d x} + \int{65536 x^{3} d x} + \int{16384 x^{5} d x} + \int{4096 x^{7} d x} + \int{1024 x^{9} d x} + \int{256 x^{11} d x} + \int{64 x^{13} d x} + \int{16 x^{15} d x} + \int{4 x^{17} d x} + \int{x^{19} d x} + \int{\frac{1048576 x}{x^{2} - 4} d x}\right)}}$$

套用冪次法則 $$$\int x^{n}\, dx = \frac{x^{n + 1}}{n + 1}$$$ $$$\left(n \neq -1 \right)$$$，以 $$$n=19$$$：

$$\int{262144 x d x} + \int{65536 x^{3} d x} + \int{16384 x^{5} d x} + \int{4096 x^{7} d x} + \int{1024 x^{9} d x} + \int{256 x^{11} d x} + \int{64 x^{13} d x} + \int{16 x^{15} d x} + \int{4 x^{17} d x} + \int{\frac{1048576 x}{x^{2} - 4} d x} + {\color{red}{\int{x^{19} d x}}}=\int{262144 x d x} + \int{65536 x^{3} d x} + \int{16384 x^{5} d x} + \int{4096 x^{7} d x} + \int{1024 x^{9} d x} + \int{256 x^{11} d x} + \int{64 x^{13} d x} + \int{16 x^{15} d x} + \int{4 x^{17} d x} + \int{\frac{1048576 x}{x^{2} - 4} d x} + {\color{red}{\frac{x^{1 + 19}}{1 + 19}}}=\int{262144 x d x} + \int{65536 x^{3} d x} + \int{16384 x^{5} d x} + \int{4096 x^{7} d x} + \int{1024 x^{9} d x} + \int{256 x^{11} d x} + \int{64 x^{13} d x} + \int{16 x^{15} d x} + \int{4 x^{17} d x} + \int{\frac{1048576 x}{x^{2} - 4} d x} + {\color{red}{\left(\frac{x^{20}}{20}\right)}}$$

套用常數倍法則 $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$，使用 $$$c=4$$$ 與 $$$f{\left(x \right)} = x^{17}$$$：

$$\frac{x^{20}}{20} + \int{262144 x d x} + \int{65536 x^{3} d x} + \int{16384 x^{5} d x} + \int{4096 x^{7} d x} + \int{1024 x^{9} d x} + \int{256 x^{11} d x} + \int{64 x^{13} d x} + \int{16 x^{15} d x} + \int{\frac{1048576 x}{x^{2} - 4} d x} + {\color{red}{\int{4 x^{17} d x}}} = \frac{x^{20}}{20} + \int{262144 x d x} + \int{65536 x^{3} d x} + \int{16384 x^{5} d x} + \int{4096 x^{7} d x} + \int{1024 x^{9} d x} + \int{256 x^{11} d x} + \int{64 x^{13} d x} + \int{16 x^{15} d x} + \int{\frac{1048576 x}{x^{2} - 4} d x} + {\color{red}{\left(4 \int{x^{17} d x}\right)}}$$

套用冪次法則 $$$\int x^{n}\, dx = \frac{x^{n + 1}}{n + 1}$$$ $$$\left(n \neq -1 \right)$$$，以 $$$n=17$$$：

$$\frac{x^{20}}{20} + \int{262144 x d x} + \int{65536 x^{3} d x} + \int{16384 x^{5} d x} + \int{4096 x^{7} d x} + \int{1024 x^{9} d x} + \int{256 x^{11} d x} + \int{64 x^{13} d x} + \int{16 x^{15} d x} + \int{\frac{1048576 x}{x^{2} - 4} d x} + 4 {\color{red}{\int{x^{17} d x}}}=\frac{x^{20}}{20} + \int{262144 x d x} + \int{65536 x^{3} d x} + \int{16384 x^{5} d x} + \int{4096 x^{7} d x} + \int{1024 x^{9} d x} + \int{256 x^{11} d x} + \int{64 x^{13} d x} + \int{16 x^{15} d x} + \int{\frac{1048576 x}{x^{2} - 4} d x} + 4 {\color{red}{\frac{x^{1 + 17}}{1 + 17}}}=\frac{x^{20}}{20} + \int{262144 x d x} + \int{65536 x^{3} d x} + \int{16384 x^{5} d x} + \int{4096 x^{7} d x} + \int{1024 x^{9} d x} + \int{256 x^{11} d x} + \int{64 x^{13} d x} + \int{16 x^{15} d x} + \int{\frac{1048576 x}{x^{2} - 4} d x} + 4 {\color{red}{\left(\frac{x^{18}}{18}\right)}}$$

套用常數倍法則 $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$，使用 $$$c=16$$$ 與 $$$f{\left(x \right)} = x^{15}$$$：

$$\frac{x^{20}}{20} + \frac{2 x^{18}}{9} + \int{262144 x d x} + \int{65536 x^{3} d x} + \int{16384 x^{5} d x} + \int{4096 x^{7} d x} + \int{1024 x^{9} d x} + \int{256 x^{11} d x} + \int{64 x^{13} d x} + \int{\frac{1048576 x}{x^{2} - 4} d x} + {\color{red}{\int{16 x^{15} d x}}} = \frac{x^{20}}{20} + \frac{2 x^{18}}{9} + \int{262144 x d x} + \int{65536 x^{3} d x} + \int{16384 x^{5} d x} + \int{4096 x^{7} d x} + \int{1024 x^{9} d x} + \int{256 x^{11} d x} + \int{64 x^{13} d x} + \int{\frac{1048576 x}{x^{2} - 4} d x} + {\color{red}{\left(16 \int{x^{15} d x}\right)}}$$

套用冪次法則 $$$\int x^{n}\, dx = \frac{x^{n + 1}}{n + 1}$$$ $$$\left(n \neq -1 \right)$$$，以 $$$n=15$$$：

$$\frac{x^{20}}{20} + \frac{2 x^{18}}{9} + \int{262144 x d x} + \int{65536 x^{3} d x} + \int{16384 x^{5} d x} + \int{4096 x^{7} d x} + \int{1024 x^{9} d x} + \int{256 x^{11} d x} + \int{64 x^{13} d x} + \int{\frac{1048576 x}{x^{2} - 4} d x} + 16 {\color{red}{\int{x^{15} d x}}}=\frac{x^{20}}{20} + \frac{2 x^{18}}{9} + \int{262144 x d x} + \int{65536 x^{3} d x} + \int{16384 x^{5} d x} + \int{4096 x^{7} d x} + \int{1024 x^{9} d x} + \int{256 x^{11} d x} + \int{64 x^{13} d x} + \int{\frac{1048576 x}{x^{2} - 4} d x} + 16 {\color{red}{\frac{x^{1 + 15}}{1 + 15}}}=\frac{x^{20}}{20} + \frac{2 x^{18}}{9} + \int{262144 x d x} + \int{65536 x^{3} d x} + \int{16384 x^{5} d x} + \int{4096 x^{7} d x} + \int{1024 x^{9} d x} + \int{256 x^{11} d x} + \int{64 x^{13} d x} + \int{\frac{1048576 x}{x^{2} - 4} d x} + 16 {\color{red}{\left(\frac{x^{16}}{16}\right)}}$$

套用常數倍法則 $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$，使用 $$$c=64$$$ 與 $$$f{\left(x \right)} = x^{13}$$$：

$$\frac{x^{20}}{20} + \frac{2 x^{18}}{9} + x^{16} + \int{262144 x d x} + \int{65536 x^{3} d x} + \int{16384 x^{5} d x} + \int{4096 x^{7} d x} + \int{1024 x^{9} d x} + \int{256 x^{11} d x} + \int{\frac{1048576 x}{x^{2} - 4} d x} + {\color{red}{\int{64 x^{13} d x}}} = \frac{x^{20}}{20} + \frac{2 x^{18}}{9} + x^{16} + \int{262144 x d x} + \int{65536 x^{3} d x} + \int{16384 x^{5} d x} + \int{4096 x^{7} d x} + \int{1024 x^{9} d x} + \int{256 x^{11} d x} + \int{\frac{1048576 x}{x^{2} - 4} d x} + {\color{red}{\left(64 \int{x^{13} d x}\right)}}$$

套用冪次法則 $$$\int x^{n}\, dx = \frac{x^{n + 1}}{n + 1}$$$ $$$\left(n \neq -1 \right)$$$，以 $$$n=13$$$：

$$\frac{x^{20}}{20} + \frac{2 x^{18}}{9} + x^{16} + \int{262144 x d x} + \int{65536 x^{3} d x} + \int{16384 x^{5} d x} + \int{4096 x^{7} d x} + \int{1024 x^{9} d x} + \int{256 x^{11} d x} + \int{\frac{1048576 x}{x^{2} - 4} d x} + 64 {\color{red}{\int{x^{13} d x}}}=\frac{x^{20}}{20} + \frac{2 x^{18}}{9} + x^{16} + \int{262144 x d x} + \int{65536 x^{3} d x} + \int{16384 x^{5} d x} + \int{4096 x^{7} d x} + \int{1024 x^{9} d x} + \int{256 x^{11} d x} + \int{\frac{1048576 x}{x^{2} - 4} d x} + 64 {\color{red}{\frac{x^{1 + 13}}{1 + 13}}}=\frac{x^{20}}{20} + \frac{2 x^{18}}{9} + x^{16} + \int{262144 x d x} + \int{65536 x^{3} d x} + \int{16384 x^{5} d x} + \int{4096 x^{7} d x} + \int{1024 x^{9} d x} + \int{256 x^{11} d x} + \int{\frac{1048576 x}{x^{2} - 4} d x} + 64 {\color{red}{\left(\frac{x^{14}}{14}\right)}}$$

套用常數倍法則 $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$，使用 $$$c=256$$$ 與 $$$f{\left(x \right)} = x^{11}$$$：

$$\frac{x^{20}}{20} + \frac{2 x^{18}}{9} + x^{16} + \frac{32 x^{14}}{7} + \int{262144 x d x} + \int{65536 x^{3} d x} + \int{16384 x^{5} d x} + \int{4096 x^{7} d x} + \int{1024 x^{9} d x} + \int{\frac{1048576 x}{x^{2} - 4} d x} + {\color{red}{\int{256 x^{11} d x}}} = \frac{x^{20}}{20} + \frac{2 x^{18}}{9} + x^{16} + \frac{32 x^{14}}{7} + \int{262144 x d x} + \int{65536 x^{3} d x} + \int{16384 x^{5} d x} + \int{4096 x^{7} d x} + \int{1024 x^{9} d x} + \int{\frac{1048576 x}{x^{2} - 4} d x} + {\color{red}{\left(256 \int{x^{11} d x}\right)}}$$

套用冪次法則 $$$\int x^{n}\, dx = \frac{x^{n + 1}}{n + 1}$$$ $$$\left(n \neq -1 \right)$$$，以 $$$n=11$$$：

$$\frac{x^{20}}{20} + \frac{2 x^{18}}{9} + x^{16} + \frac{32 x^{14}}{7} + \int{262144 x d x} + \int{65536 x^{3} d x} + \int{16384 x^{5} d x} + \int{4096 x^{7} d x} + \int{1024 x^{9} d x} + \int{\frac{1048576 x}{x^{2} - 4} d x} + 256 {\color{red}{\int{x^{11} d x}}}=\frac{x^{20}}{20} + \frac{2 x^{18}}{9} + x^{16} + \frac{32 x^{14}}{7} + \int{262144 x d x} + \int{65536 x^{3} d x} + \int{16384 x^{5} d x} + \int{4096 x^{7} d x} + \int{1024 x^{9} d x} + \int{\frac{1048576 x}{x^{2} - 4} d x} + 256 {\color{red}{\frac{x^{1 + 11}}{1 + 11}}}=\frac{x^{20}}{20} + \frac{2 x^{18}}{9} + x^{16} + \frac{32 x^{14}}{7} + \int{262144 x d x} + \int{65536 x^{3} d x} + \int{16384 x^{5} d x} + \int{4096 x^{7} d x} + \int{1024 x^{9} d x} + \int{\frac{1048576 x}{x^{2} - 4} d x} + 256 {\color{red}{\left(\frac{x^{12}}{12}\right)}}$$

套用常數倍法則 $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$，使用 $$$c=1024$$$ 與 $$$f{\left(x \right)} = x^{9}$$$：

$$\frac{x^{20}}{20} + \frac{2 x^{18}}{9} + x^{16} + \frac{32 x^{14}}{7} + \frac{64 x^{12}}{3} + \int{262144 x d x} + \int{65536 x^{3} d x} + \int{16384 x^{5} d x} + \int{4096 x^{7} d x} + \int{\frac{1048576 x}{x^{2} - 4} d x} + {\color{red}{\int{1024 x^{9} d x}}} = \frac{x^{20}}{20} + \frac{2 x^{18}}{9} + x^{16} + \frac{32 x^{14}}{7} + \frac{64 x^{12}}{3} + \int{262144 x d x} + \int{65536 x^{3} d x} + \int{16384 x^{5} d x} + \int{4096 x^{7} d x} + \int{\frac{1048576 x}{x^{2} - 4} d x} + {\color{red}{\left(1024 \int{x^{9} d x}\right)}}$$

套用冪次法則 $$$\int x^{n}\, dx = \frac{x^{n + 1}}{n + 1}$$$ $$$\left(n \neq -1 \right)$$$，以 $$$n=9$$$：

$$\frac{x^{20}}{20} + \frac{2 x^{18}}{9} + x^{16} + \frac{32 x^{14}}{7} + \frac{64 x^{12}}{3} + \int{262144 x d x} + \int{65536 x^{3} d x} + \int{16384 x^{5} d x} + \int{4096 x^{7} d x} + \int{\frac{1048576 x}{x^{2} - 4} d x} + 1024 {\color{red}{\int{x^{9} d x}}}=\frac{x^{20}}{20} + \frac{2 x^{18}}{9} + x^{16} + \frac{32 x^{14}}{7} + \frac{64 x^{12}}{3} + \int{262144 x d x} + \int{65536 x^{3} d x} + \int{16384 x^{5} d x} + \int{4096 x^{7} d x} + \int{\frac{1048576 x}{x^{2} - 4} d x} + 1024 {\color{red}{\frac{x^{1 + 9}}{1 + 9}}}=\frac{x^{20}}{20} + \frac{2 x^{18}}{9} + x^{16} + \frac{32 x^{14}}{7} + \frac{64 x^{12}}{3} + \int{262144 x d x} + \int{65536 x^{3} d x} + \int{16384 x^{5} d x} + \int{4096 x^{7} d x} + \int{\frac{1048576 x}{x^{2} - 4} d x} + 1024 {\color{red}{\left(\frac{x^{10}}{10}\right)}}$$

套用常數倍法則 $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$，使用 $$$c=4096$$$ 與 $$$f{\left(x \right)} = x^{7}$$$：

$$\frac{x^{20}}{20} + \frac{2 x^{18}}{9} + x^{16} + \frac{32 x^{14}}{7} + \frac{64 x^{12}}{3} + \frac{512 x^{10}}{5} + \int{262144 x d x} + \int{65536 x^{3} d x} + \int{16384 x^{5} d x} + \int{\frac{1048576 x}{x^{2} - 4} d x} + {\color{red}{\int{4096 x^{7} d x}}} = \frac{x^{20}}{20} + \frac{2 x^{18}}{9} + x^{16} + \frac{32 x^{14}}{7} + \frac{64 x^{12}}{3} + \frac{512 x^{10}}{5} + \int{262144 x d x} + \int{65536 x^{3} d x} + \int{16384 x^{5} d x} + \int{\frac{1048576 x}{x^{2} - 4} d x} + {\color{red}{\left(4096 \int{x^{7} d x}\right)}}$$

套用冪次法則 $$$\int x^{n}\, dx = \frac{x^{n + 1}}{n + 1}$$$ $$$\left(n \neq -1 \right)$$$，以 $$$n=7$$$：

$$\frac{x^{20}}{20} + \frac{2 x^{18}}{9} + x^{16} + \frac{32 x^{14}}{7} + \frac{64 x^{12}}{3} + \frac{512 x^{10}}{5} + \int{262144 x d x} + \int{65536 x^{3} d x} + \int{16384 x^{5} d x} + \int{\frac{1048576 x}{x^{2} - 4} d x} + 4096 {\color{red}{\int{x^{7} d x}}}=\frac{x^{20}}{20} + \frac{2 x^{18}}{9} + x^{16} + \frac{32 x^{14}}{7} + \frac{64 x^{12}}{3} + \frac{512 x^{10}}{5} + \int{262144 x d x} + \int{65536 x^{3} d x} + \int{16384 x^{5} d x} + \int{\frac{1048576 x}{x^{2} - 4} d x} + 4096 {\color{red}{\frac{x^{1 + 7}}{1 + 7}}}=\frac{x^{20}}{20} + \frac{2 x^{18}}{9} + x^{16} + \frac{32 x^{14}}{7} + \frac{64 x^{12}}{3} + \frac{512 x^{10}}{5} + \int{262144 x d x} + \int{65536 x^{3} d x} + \int{16384 x^{5} d x} + \int{\frac{1048576 x}{x^{2} - 4} d x} + 4096 {\color{red}{\left(\frac{x^{8}}{8}\right)}}$$

套用常數倍法則 $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$，使用 $$$c=16384$$$ 與 $$$f{\left(x \right)} = x^{5}$$$：

$$\frac{x^{20}}{20} + \frac{2 x^{18}}{9} + x^{16} + \frac{32 x^{14}}{7} + \frac{64 x^{12}}{3} + \frac{512 x^{10}}{5} + 512 x^{8} + \int{262144 x d x} + \int{65536 x^{3} d x} + \int{\frac{1048576 x}{x^{2} - 4} d x} + {\color{red}{\int{16384 x^{5} d x}}} = \frac{x^{20}}{20} + \frac{2 x^{18}}{9} + x^{16} + \frac{32 x^{14}}{7} + \frac{64 x^{12}}{3} + \frac{512 x^{10}}{5} + 512 x^{8} + \int{262144 x d x} + \int{65536 x^{3} d x} + \int{\frac{1048576 x}{x^{2} - 4} d x} + {\color{red}{\left(16384 \int{x^{5} d x}\right)}}$$

套用冪次法則 $$$\int x^{n}\, dx = \frac{x^{n + 1}}{n + 1}$$$ $$$\left(n \neq -1 \right)$$$，以 $$$n=5$$$：

$$\frac{x^{20}}{20} + \frac{2 x^{18}}{9} + x^{16} + \frac{32 x^{14}}{7} + \frac{64 x^{12}}{3} + \frac{512 x^{10}}{5} + 512 x^{8} + \int{262144 x d x} + \int{65536 x^{3} d x} + \int{\frac{1048576 x}{x^{2} - 4} d x} + 16384 {\color{red}{\int{x^{5} d x}}}=\frac{x^{20}}{20} + \frac{2 x^{18}}{9} + x^{16} + \frac{32 x^{14}}{7} + \frac{64 x^{12}}{3} + \frac{512 x^{10}}{5} + 512 x^{8} + \int{262144 x d x} + \int{65536 x^{3} d x} + \int{\frac{1048576 x}{x^{2} - 4} d x} + 16384 {\color{red}{\frac{x^{1 + 5}}{1 + 5}}}=\frac{x^{20}}{20} + \frac{2 x^{18}}{9} + x^{16} + \frac{32 x^{14}}{7} + \frac{64 x^{12}}{3} + \frac{512 x^{10}}{5} + 512 x^{8} + \int{262144 x d x} + \int{65536 x^{3} d x} + \int{\frac{1048576 x}{x^{2} - 4} d x} + 16384 {\color{red}{\left(\frac{x^{6}}{6}\right)}}$$

套用常數倍法則 $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$，使用 $$$c=65536$$$ 與 $$$f{\left(x \right)} = x^{3}$$$：

$$\frac{x^{20}}{20} + \frac{2 x^{18}}{9} + x^{16} + \frac{32 x^{14}}{7} + \frac{64 x^{12}}{3} + \frac{512 x^{10}}{5} + 512 x^{8} + \frac{8192 x^{6}}{3} + \int{262144 x d x} + \int{\frac{1048576 x}{x^{2} - 4} d x} + {\color{red}{\int{65536 x^{3} d x}}} = \frac{x^{20}}{20} + \frac{2 x^{18}}{9} + x^{16} + \frac{32 x^{14}}{7} + \frac{64 x^{12}}{3} + \frac{512 x^{10}}{5} + 512 x^{8} + \frac{8192 x^{6}}{3} + \int{262144 x d x} + \int{\frac{1048576 x}{x^{2} - 4} d x} + {\color{red}{\left(65536 \int{x^{3} d x}\right)}}$$

套用冪次法則 $$$\int x^{n}\, dx = \frac{x^{n + 1}}{n + 1}$$$ $$$\left(n \neq -1 \right)$$$，以 $$$n=3$$$：

$$\frac{x^{20}}{20} + \frac{2 x^{18}}{9} + x^{16} + \frac{32 x^{14}}{7} + \frac{64 x^{12}}{3} + \frac{512 x^{10}}{5} + 512 x^{8} + \frac{8192 x^{6}}{3} + \int{262144 x d x} + \int{\frac{1048576 x}{x^{2} - 4} d x} + 65536 {\color{red}{\int{x^{3} d x}}}=\frac{x^{20}}{20} + \frac{2 x^{18}}{9} + x^{16} + \frac{32 x^{14}}{7} + \frac{64 x^{12}}{3} + \frac{512 x^{10}}{5} + 512 x^{8} + \frac{8192 x^{6}}{3} + \int{262144 x d x} + \int{\frac{1048576 x}{x^{2} - 4} d x} + 65536 {\color{red}{\frac{x^{1 + 3}}{1 + 3}}}=\frac{x^{20}}{20} + \frac{2 x^{18}}{9} + x^{16} + \frac{32 x^{14}}{7} + \frac{64 x^{12}}{3} + \frac{512 x^{10}}{5} + 512 x^{8} + \frac{8192 x^{6}}{3} + \int{262144 x d x} + \int{\frac{1048576 x}{x^{2} - 4} d x} + 65536 {\color{red}{\left(\frac{x^{4}}{4}\right)}}$$

套用常數倍法則 $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$，使用 $$$c=262144$$$ 與 $$$f{\left(x \right)} = x$$$：

$$\frac{x^{20}}{20} + \frac{2 x^{18}}{9} + x^{16} + \frac{32 x^{14}}{7} + \frac{64 x^{12}}{3} + \frac{512 x^{10}}{5} + 512 x^{8} + \frac{8192 x^{6}}{3} + 16384 x^{4} + \int{\frac{1048576 x}{x^{2} - 4} d x} + {\color{red}{\int{262144 x d x}}} = \frac{x^{20}}{20} + \frac{2 x^{18}}{9} + x^{16} + \frac{32 x^{14}}{7} + \frac{64 x^{12}}{3} + \frac{512 x^{10}}{5} + 512 x^{8} + \frac{8192 x^{6}}{3} + 16384 x^{4} + \int{\frac{1048576 x}{x^{2} - 4} d x} + {\color{red}{\left(262144 \int{x d x}\right)}}$$

套用冪次法則 $$$\int x^{n}\, dx = \frac{x^{n + 1}}{n + 1}$$$ $$$\left(n \neq -1 \right)$$$，以 $$$n=1$$$：

$$\frac{x^{20}}{20} + \frac{2 x^{18}}{9} + x^{16} + \frac{32 x^{14}}{7} + \frac{64 x^{12}}{3} + \frac{512 x^{10}}{5} + 512 x^{8} + \frac{8192 x^{6}}{3} + 16384 x^{4} + \int{\frac{1048576 x}{x^{2} - 4} d x} + 262144 {\color{red}{\int{x d x}}}=\frac{x^{20}}{20} + \frac{2 x^{18}}{9} + x^{16} + \frac{32 x^{14}}{7} + \frac{64 x^{12}}{3} + \frac{512 x^{10}}{5} + 512 x^{8} + \frac{8192 x^{6}}{3} + 16384 x^{4} + \int{\frac{1048576 x}{x^{2} - 4} d x} + 262144 {\color{red}{\frac{x^{1 + 1}}{1 + 1}}}=\frac{x^{20}}{20} + \frac{2 x^{18}}{9} + x^{16} + \frac{32 x^{14}}{7} + \frac{64 x^{12}}{3} + \frac{512 x^{10}}{5} + 512 x^{8} + \frac{8192 x^{6}}{3} + 16384 x^{4} + \int{\frac{1048576 x}{x^{2} - 4} d x} + 262144 {\color{red}{\left(\frac{x^{2}}{2}\right)}}$$

令 $$$u=x^{2} - 4$$$。

則 $$$du=\left(x^{2} - 4\right)^{\prime }dx = 2 x dx$$$ (步驟見»)，並可得 $$$x dx = \frac{du}{2}$$$。

因此，

$$\frac{x^{20}}{20} + \frac{2 x^{18}}{9} + x^{16} + \frac{32 x^{14}}{7} + \frac{64 x^{12}}{3} + \frac{512 x^{10}}{5} + 512 x^{8} + \frac{8192 x^{6}}{3} + 16384 x^{4} + 131072 x^{2} + {\color{red}{\int{\frac{1048576 x}{x^{2} - 4} d x}}} = \frac{x^{20}}{20} + \frac{2 x^{18}}{9} + x^{16} + \frac{32 x^{14}}{7} + \frac{64 x^{12}}{3} + \frac{512 x^{10}}{5} + 512 x^{8} + \frac{8192 x^{6}}{3} + 16384 x^{4} + 131072 x^{2} + {\color{red}{\int{\frac{524288}{u} d u}}}$$

套用常數倍法則 $$$\int c f{\left(u \right)}\, du = c \int f{\left(u \right)}\, du$$$，使用 $$$c=524288$$$ 與 $$$f{\left(u \right)} = \frac{1}{u}$$$：

$$\frac{x^{20}}{20} + \frac{2 x^{18}}{9} + x^{16} + \frac{32 x^{14}}{7} + \frac{64 x^{12}}{3} + \frac{512 x^{10}}{5} + 512 x^{8} + \frac{8192 x^{6}}{3} + 16384 x^{4} + 131072 x^{2} + {\color{red}{\int{\frac{524288}{u} d u}}} = \frac{x^{20}}{20} + \frac{2 x^{18}}{9} + x^{16} + \frac{32 x^{14}}{7} + \frac{64 x^{12}}{3} + \frac{512 x^{10}}{5} + 512 x^{8} + \frac{8192 x^{6}}{3} + 16384 x^{4} + 131072 x^{2} + {\color{red}{\left(524288 \int{\frac{1}{u} d u}\right)}}$$

$$$\frac{1}{u}$$$ 的積分是 $$$\int{\frac{1}{u} d u} = \ln{\left(\left|{u}\right| \right)}$$$：

$$\frac{x^{20}}{20} + \frac{2 x^{18}}{9} + x^{16} + \frac{32 x^{14}}{7} + \frac{64 x^{12}}{3} + \frac{512 x^{10}}{5} + 512 x^{8} + \frac{8192 x^{6}}{3} + 16384 x^{4} + 131072 x^{2} + 524288 {\color{red}{\int{\frac{1}{u} d u}}} = \frac{x^{20}}{20} + \frac{2 x^{18}}{9} + x^{16} + \frac{32 x^{14}}{7} + \frac{64 x^{12}}{3} + \frac{512 x^{10}}{5} + 512 x^{8} + \frac{8192 x^{6}}{3} + 16384 x^{4} + 131072 x^{2} + 524288 {\color{red}{\ln{\left(\left|{u}\right| \right)}}}$$

回顧一下 $$$u=x^{2} - 4$$$：

$$\frac{x^{20}}{20} + \frac{2 x^{18}}{9} + x^{16} + \frac{32 x^{14}}{7} + \frac{64 x^{12}}{3} + \frac{512 x^{10}}{5} + 512 x^{8} + \frac{8192 x^{6}}{3} + 16384 x^{4} + 131072 x^{2} + 524288 \ln{\left(\left|{{\color{red}{u}}}\right| \right)} = \frac{x^{20}}{20} + \frac{2 x^{18}}{9} + x^{16} + \frac{32 x^{14}}{7} + \frac{64 x^{12}}{3} + \frac{512 x^{10}}{5} + 512 x^{8} + \frac{8192 x^{6}}{3} + 16384 x^{4} + 131072 x^{2} + 524288 \ln{\left(\left|{{\color{red}{\left(x^{2} - 4\right)}}}\right| \right)}$$

因此，

$$\int{\frac{x^{21}}{x^{2} - 4} d x} = \frac{x^{20}}{20} + \frac{2 x^{18}}{9} + x^{16} + \frac{32 x^{14}}{7} + \frac{64 x^{12}}{3} + \frac{512 x^{10}}{5} + 512 x^{8} + \frac{8192 x^{6}}{3} + 16384 x^{4} + 131072 x^{2} + 524288 \ln{\left(\left|{x^{2} - 4}\right| \right)}$$

加上積分常數：

$$\int{\frac{x^{21}}{x^{2} - 4} d x} = \frac{x^{20}}{20} + \frac{2 x^{18}}{9} + x^{16} + \frac{32 x^{14}}{7} + \frac{64 x^{12}}{3} + \frac{512 x^{10}}{5} + 512 x^{8} + \frac{8192 x^{6}}{3} + 16384 x^{4} + 131072 x^{2} + 524288 \ln{\left(\left|{x^{2} - 4}\right| \right)}+C$$

$$$\int \frac{x^{21}}{x^{2} - 4}\, dx = \left(\frac{x^{20}}{20} + \frac{2 x^{18}}{9} + x^{16} + \frac{32 x^{14}}{7} + \frac{64 x^{12}}{3} + \frac{512 x^{10}}{5} + 512 x^{8} + \frac{8192 x^{6}}{3} + 16384 x^{4} + 131072 x^{2} + 524288 \ln\left(\left|{x^{2} - 4}\right|\right)\right) + C$$$A