What is the integral of (x-1/(x^2))(x^2+2/x)^9 ?

The integral of (x-1/(x^2))(x^2+2/x)^9 is

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integral of (x-1/(x^2))(x^2+2/x)^9

\int (x - \frac{1}{x ^{2}}) (x^{2} + \frac{2}{x})^{9} d x

\frac{x ^{20}}{20} + x^{17} + 9 x^{14} + 48 x^{11} + 168 x^{8} + \frac{2016 x ^{5}}{5} + 672 x^{2} + \frac{768}{x} + \frac{576}{x ^{4}} + \frac{256}{x ^{7}} + \frac{256}{5 x ^{10}} + C

Solution steps

\int (x - \frac{1}{x ^{2}}) (x^{2} + \frac{2}{x})^{9} d x

Expand

(x - \frac{1}{x ^{2}}) (x^{2} + \frac{2}{x})^{9} : x^{19} + 17 x^{16} + 126 x^{13} + 528 x^{10} + 1344 x^{7} + 2016 x^{4} + 1344 x - \frac{768}{x ^{2}} - \frac{2304}{x ^{5}} - \frac{1792}{x ^{8}} - \frac{512}{x ^{11}}

= \int x^{19} + 17 x^{16} + 126 x^{13} + 528 x^{10} + 1344 x^{7} + 2016 x^{4} + 1344 x - \frac{768}{x ^{2}} - \frac{2304}{x ^{5}} - \frac{1792}{x ^{8}} - \frac{512}{x ^{11}} d x

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \int x^{19} d x + \int 17 x^{16} d x + \int 126 x^{13} d x + \int 528 x^{10} d x + \int 1344 x^{7} d x + \int 2016 x^{4} d x + \int 1344 x d x - \int \frac{768}{x ^{2}} d x - \int \frac{2304}{x ^{5}} d x - \int \frac{1792}{x ^{8}} d x - \int \frac{512}{x ^{11}} d x

\int x^{19} d x = \frac{x ^{20}}{20}

\int 17 x^{16} d x = x^{17}

\int 126 x^{13} d x = 9 x^{14}

\int 528 x^{10} d x = 48 x^{11}

\int 1344 x^{7} d x = 168 x^{8}

\int 2016 x^{4} d x = \frac{2016 x ^{5}}{5}

\int 1344 x d x = 672 x^{2}

\int \frac{768}{x ^{2}} d x = - \frac{768}{x}

\int \frac{2304}{x ^{5}} d x = - \frac{576}{x ^{4}}

\int \frac{1792}{x ^{8}} d x = - \frac{256}{x ^{7}}

\int \frac{512}{x ^{11}} d x = - \frac{256}{5 x ^{10}}

= \frac{x ^{20}}{20} + x^{17} + 9 x^{14} + 48 x^{11} + 168 x^{8} + \frac{2016 x ^{5}}{5} + 672 x^{2} - (- \frac{768}{x}) - (- \frac{576}{x ^{4}}) - (- \frac{256}{x ^{7}}) - (- \frac{256}{5 x ^{10}})

Simplify

= \frac{x ^{20}}{20} + x^{17} + 9 x^{14} + 48 x^{11} + 168 x^{8} + \frac{2016 x ^{5}}{5} + 672 x^{2} + \frac{768}{x} + \frac{576}{x ^{4}} + \frac{256}{x ^{7}} + \frac{256}{5 x ^{10}}

Add a constant to the solution

= \frac{x ^{20}}{20} + x^{17} + 9 x^{14} + 48 x^{11} + 168 x^{8} + \frac{2016 x ^{5}}{5} + 672 x^{2} + \frac{768}{x} + \frac{576}{x ^{4}} + \frac{256}{x ^{7}} + \frac{256}{5 x ^{10}} + C

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What is the integral of (x-1/(x^2))(x^2+2/x)^9 ?
The integral of (x-1/(x^2))(x^2+2/x)^9 is