What is the integral of ((x^{10}))/(x^2+x-2) ?

The integral of ((x^{10}))/(x^2+x-2) is

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integral of ((x^{10}))/(x^2+x-2)

\int \frac{( x ^{10} )}{x ^{2} + x - 2} d x

\frac{x ^{9}}{9} - \frac{x ^{8}}{8} + \frac{3 x ^{7}}{7} - \frac{5 x ^{6}}{6} + \frac{11 x ^{5}}{5} - \frac{21 x ^{4}}{4} + \frac{43 x ^{3}}{3} - \frac{85 x ^{2}}{2} + 171 x + \frac{1}{3} ln ∣ x - 1 ∣ - \frac{1024}{3} ln ∣ x + 2 ∣ + C

Solution steps

\int \frac{x ^{10}}{x ^{2} + x - 2} d x

Long division

\frac{x ^{10}}{x ^{2} + x - 2} : x^{8} - x^{7} + 3 x^{6} - 5 x^{5} + 11 x^{4} - 21 x^{3} + 43 x^{2} - 85 x + 171 + \frac{- 341 x + 342}{x ^{2} + x - 2}

= \int x^{8} - x^{7} + 3 x^{6} - 5 x^{5} + 11 x^{4} - 21 x^{3} + 43 x^{2} - 85 x + 171 + \frac{- 341 x + 342}{x ^{2} + x - 2} 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^{8} d x - \int x^{7} d x + \int 3 x^{6} d x - \int 5 x^{5} d x + \int 11 x^{4} d x - \int 21 x^{3} d x + \int 43 x^{2} d x - \int 85 x d x + \int 171 d x + \int \frac{- 341 x + 342}{x ^{2} + x - 2} d x

\int x^{8} d x = \frac{x ^{9}}{9}

\int x^{7} d x = \frac{x ^{8}}{8}

\int 3 x^{6} d x = \frac{3 x ^{7}}{7}

\int 5 x^{5} d x = \frac{5 x ^{6}}{6}

\int 11 x^{4} d x = \frac{11 x ^{5}}{5}

\int 21 x^{3} d x = \frac{21 x ^{4}}{4}

\int 43 x^{2} d x = \frac{43 x ^{3}}{3}

\int 85 x d x = \frac{85 x ^{2}}{2}

\int 171 d x = 171 x

\int \frac{- 341 x + 342}{x ^{2} + x - 2} d x = \frac{1}{3} ln ∣ x - 1 ∣ - \frac{1024}{3} ln ∣ x + 2 ∣

= \frac{x ^{9}}{9} - \frac{x ^{8}}{8} + \frac{3 x ^{7}}{7} - \frac{5 x ^{6}}{6} + \frac{11 x ^{5}}{5} - \frac{21 x ^{4}}{4} + \frac{43 x ^{3}}{3} - \frac{85 x ^{2}}{2} + 171 x + \frac{1}{3} ln ∣ x - 1 ∣ - \frac{1024}{3} ln ∣ x + 2 ∣

Add a constant to the solution

= \frac{x ^{9}}{9} - \frac{x ^{8}}{8} + \frac{3 x ^{7}}{7} - \frac{5 x ^{6}}{6} + \frac{11 x ^{5}}{5} - \frac{21 x ^{4}}{4} + \frac{43 x ^{3}}{3} - \frac{85 x ^{2}}{2} + 171 x + \frac{1}{3} ln ∣ x - 1 ∣ - \frac{1024}{3} ln ∣ x + 2 ∣ + C

f (x) = \frac{4}{x}, at x = 3

\frac{1}{4 s ^{2} + 1}

x \to 0 lim (x^{2} cos (\frac{2}{x}))

x \to - 4 lim (∣ x + 4 ∣)

\frac{2 x}{( x + 1 )}

What is the integral of ((x^{10}))/(x^2+x-2) ?
The integral of ((x^{10}))/(x^2+x-2) is