What is the integral of (x^7)/((1-x^2)^5) ?

The integral of (x^7)/((1-x^2)^5) is

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integral of (x^7)/((1-x^2)^5)

\int \frac{x ^{7}}{( 1 - x ^{2} ) ^{5}} d x

- \frac{35}{256 ( x + 1 )} + \frac{29}{256 ( x + 1 ) ^{2}} - \frac{3}{64 ( x + 1 ) ^{3}} + \frac{1}{128 ( x + 1 ) ^{4}} + \frac{35}{256 ( x - 1 )} + \frac{29}{256 ( x - 1 ) ^{2}} + \frac{3}{64 ( x - 1 ) ^{3}} + \frac{1}{128 ( x - 1 ) ^{4}} + C

Solution steps

\int \frac{x ^{7}}{( 1 - x ^{2} ) ^{5}} d x

Take the partial fraction of

\frac{x ^{7}}{( 1 - x ^{2} ) ^{5}} : \frac{35}{256 ( x + 1 ) ^{2}} - \frac{29}{128 ( x + 1 ) ^{3}} + \frac{9}{64 ( x + 1 ) ^{4}} - \frac{1}{32 ( x + 1 ) ^{5}} - \frac{35}{256 ( x - 1 ) ^{2}} - \frac{29}{128 ( x - 1 ) ^{3}} - \frac{9}{64 ( x - 1 ) ^{4}} - \frac{1}{32 ( x - 1 ) ^{5}}

= \int \frac{35}{256 ( x + 1 ) ^{2}} - \frac{29}{128 ( x + 1 ) ^{3}} + \frac{9}{64 ( x + 1 ) ^{4}} - \frac{1}{32 ( x + 1 ) ^{5}} - \frac{35}{256 ( x - 1 ) ^{2}} - \frac{29}{128 ( x - 1 ) ^{3}} - \frac{9}{64 ( x - 1 ) ^{4}} - \frac{1}{32 ( x - 1 ) ^{5}} 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 \frac{35}{256 ( x + 1 ) ^{2}} d x - \int \frac{29}{128 ( x + 1 ) ^{3}} d x + \int \frac{9}{64 ( x + 1 ) ^{4}} d x - \int \frac{1}{32 ( x + 1 ) ^{5}} d x - \int \frac{35}{256 ( x - 1 ) ^{2}} d x - \int \frac{29}{128 ( x - 1 ) ^{3}} d x - \int \frac{9}{64 ( x - 1 ) ^{4}} d x - \int \frac{1}{32 ( x - 1 ) ^{5}} d x

\int \frac{35}{256 ( x + 1 ) ^{2}} d x = - \frac{35}{256 ( x + 1 )}

\int \frac{29}{128 ( x + 1 ) ^{3}} d x = - \frac{29}{256 ( x + 1 ) ^{2}}

\int \frac{9}{64 ( x + 1 ) ^{4}} d x = - \frac{3}{64 ( x + 1 ) ^{3}}

\int \frac{1}{32 ( x + 1 ) ^{5}} d x = - \frac{1}{128 ( x + 1 ) ^{4}}

\int \frac{35}{256 ( x - 1 ) ^{2}} d x = - \frac{35}{256 ( x - 1 )}

\int \frac{29}{128 ( x - 1 ) ^{3}} d x = - \frac{29}{256 ( x - 1 ) ^{2}}

\int \frac{9}{64 ( x - 1 ) ^{4}} d x = - \frac{3}{64 ( x - 1 ) ^{3}}

\int \frac{1}{32 ( x - 1 ) ^{5}} d x = - \frac{1}{128 ( x - 1 ) ^{4}}

= - \frac{35}{256 ( x + 1 )} - (- \frac{29}{256 ( x + 1 ) ^{2}}) - \frac{3}{64 ( x + 1 ) ^{3}} - (- \frac{1}{128 ( x + 1 ) ^{4}}) - (- \frac{35}{256 ( x - 1 )}) - (- \frac{29}{256 ( x - 1 ) ^{2}}) - (- \frac{3}{64 ( x - 1 ) ^{3}}) - (- \frac{1}{128 ( x - 1 ) ^{4}})

Simplify

= - \frac{35}{256 ( x + 1 )} + \frac{29}{256 ( x + 1 ) ^{2}} - \frac{3}{64 ( x + 1 ) ^{3}} + \frac{1}{128 ( x + 1 ) ^{4}} + \frac{35}{256 ( x - 1 )} + \frac{29}{256 ( x - 1 ) ^{2}} + \frac{3}{64 ( x - 1 ) ^{3}} + \frac{1}{128 ( x - 1 ) ^{4}}

Add a constant to the solution

= - \frac{35}{256 ( x + 1 )} + \frac{29}{256 ( x + 1 ) ^{2}} - \frac{3}{64 ( x + 1 ) ^{3}} + \frac{1}{128 ( x + 1 ) ^{4}} + \frac{35}{256 ( x - 1 )} + \frac{29}{256 ( x - 1 ) ^{2}} + \frac{3}{64 ( x - 1 ) ^{3}} + \frac{1}{128 ( x - 1 ) ^{4}} + C

What is the integral of (x^7)/((1-x^2)^5) ?
The integral of (x^7)/((1-x^2)^5) is