Integral of $$$\frac{x^{21}}{x^{2} - 4}$$$

Find $$$\int \frac{x^{21}}{x^{2} - 4}\, dx$$$.

Since the degree of the numerator is not less than the degree of the denominator, perform polynomial long division (steps can be seen »):

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

Integrate term by term:

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

Apply the power rule $$$\int x^{n}\, dx = \frac{x^{n + 1}}{n + 1}$$$ $$$\left(n \neq -1 \right)$$$ with $$$n=19$$$:

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

Apply the constant multiple rule $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ with $$$c=4$$$ and $$$f{\left(x \right)} = x^{17}$$$:

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

Apply the power rule $$$\int x^{n}\, dx = \frac{x^{n + 1}}{n + 1}$$$ $$$\left(n \neq -1 \right)$$$ with $$$n=17$$$:

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

Apply the constant multiple rule $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ with $$$c=16$$$ and $$$f{\left(x \right)} = x^{15}$$$:

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

Apply the power rule $$$\int x^{n}\, dx = \frac{x^{n + 1}}{n + 1}$$$ $$$\left(n \neq -1 \right)$$$ with $$$n=15$$$:

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

Apply the constant multiple rule $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ with $$$c=64$$$ and $$$f{\left(x \right)} = x^{13}$$$:

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

Apply the power rule $$$\int x^{n}\, dx = \frac{x^{n + 1}}{n + 1}$$$ $$$\left(n \neq -1 \right)$$$ with $$$n=13$$$:

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

Apply the constant multiple rule $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ with $$$c=256$$$ and $$$f{\left(x \right)} = x^{11}$$$:

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

Apply the power rule $$$\int x^{n}\, dx = \frac{x^{n + 1}}{n + 1}$$$ $$$\left(n \neq -1 \right)$$$ with $$$n=11$$$:

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

Apply the constant multiple rule $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ with $$$c=1024$$$ and $$$f{\left(x \right)} = x^{9}$$$:

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

Apply the power rule $$$\int x^{n}\, dx = \frac{x^{n + 1}}{n + 1}$$$ $$$\left(n \neq -1 \right)$$$ with $$$n=9$$$:

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

Apply the constant multiple rule $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ with $$$c=4096$$$ and $$$f{\left(x \right)} = x^{7}$$$:

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

Apply the power rule $$$\int x^{n}\, dx = \frac{x^{n + 1}}{n + 1}$$$ $$$\left(n \neq -1 \right)$$$ with $$$n=7$$$:

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

Apply the constant multiple rule $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ with $$$c=16384$$$ and $$$f{\left(x \right)} = x^{5}$$$:

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

Apply the power rule $$$\int x^{n}\, dx = \frac{x^{n + 1}}{n + 1}$$$ $$$\left(n \neq -1 \right)$$$ with $$$n=5$$$:

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

Apply the constant multiple rule $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ with $$$c=65536$$$ and $$$f{\left(x \right)} = x^{3}$$$:

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

Apply the power rule $$$\int x^{n}\, dx = \frac{x^{n + 1}}{n + 1}$$$ $$$\left(n \neq -1 \right)$$$ with $$$n=3$$$:

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

Apply the constant multiple rule $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ with $$$c=262144$$$ and $$$f{\left(x \right)} = x$$$:

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

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

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

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

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

The integral can be rewritten as

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

Apply the constant multiple rule $$$\int c f{\left(u \right)}\, du = c \int f{\left(u \right)}\, du$$$ with $$$c=524288$$$ and $$$f{\left(u \right)} = \frac{1}{u}$$$:

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

The integral of $$$\frac{1}{u}$$$ is $$$\int{\frac{1}{u} d u} = \ln{\left(\left|{u}\right| \right)}$$$:

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

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

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

Therefore,

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

Add the constant of integration:

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

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