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

解

\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

解答ステップ

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

長除法

\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

総和規則を適用する:

\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 ∣

定数を解答に追加する

= \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