integral of 1/((x^4-1)^2)

解

\int \frac{1}{( x ^{4} - 1 ) ^{2}} d x

\frac{3}{8} arctan (x) + \frac{x}{8 ( 1 + x ^{2} )} + \frac{3}{16} ln ∣ x + 1 ∣ - \frac{1}{16 ( x + 1 )} - \frac{3}{16} ln ∣ x - 1 ∣ - \frac{1}{16 ( x - 1 )} + C

解答ステップ

\int \frac{1}{( x ^{4} - 1 ) ^{2}} d x

以下の部分分数を得る:

\frac{1}{( x ^{4} - 1 ) ^{2}} : \frac{1}{4 ( x ^{2} + 1 )} + \frac{1}{4 ( x ^{2} + 1 ) ^{2}} + \frac{3}{16 ( x + 1 )} + \frac{1}{16 ( x + 1 ) ^{2}} - \frac{3}{16 ( x - 1 )} + \frac{1}{16 ( x - 1 ) ^{2}}

= \int \frac{1}{4 ( x ^{2} + 1 )} + \frac{1}{4 ( x ^{2} + 1 ) ^{2}} + \frac{3}{16 ( x + 1 )} + \frac{1}{16 ( x + 1 ) ^{2}} - \frac{3}{16 ( x - 1 )} + \frac{1}{16 ( x - 1 ) ^{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 \frac{1}{4 ( x ^{2} + 1 )} d x + \int \frac{1}{4 ( x ^{2} + 1 ) ^{2}} d x + \int \frac{3}{16 ( x + 1 )} d x + \int \frac{1}{16 ( x + 1 ) ^{2}} d x - \int \frac{3}{16 ( x - 1 )} d x + \int \frac{1}{16 ( x - 1 ) ^{2}} d x

\int \frac{1}{4 ( x ^{2} + 1 )} d x = \frac{1}{4} arctan (x)

\int \frac{1}{4 ( x ^{2} + 1 ) ^{2}} d x = \frac{1}{8} (arctan (x) + \frac{x}{1 + x ^{2}})

\int \frac{3}{16 ( x + 1 )} d x = \frac{3}{16} ln ∣ x + 1 ∣

\int \frac{1}{16 ( x + 1 ) ^{2}} d x = - \frac{1}{16 ( x + 1 )}

\int \frac{3}{16 ( x - 1 )} d x = \frac{3}{16} ln ∣ x - 1 ∣

\int \frac{1}{16 ( x - 1 ) ^{2}} d x = - \frac{1}{16 ( x - 1 )}

= \frac{1}{4} arctan (x) + \frac{1}{8} (arctan (x) + \frac{x}{1 + x ^{2}}) + \frac{3}{16} ln ∣ x + 1 ∣ - \frac{1}{16 ( x + 1 )} - \frac{3}{16} ln ∣ x - 1 ∣ - \frac{1}{16 ( x - 1 )}

簡素化

\frac{1}{4} arctan (x) + \frac{1}{8} (arctan (x) + \frac{x}{1 + x ^{2}}) + \frac{3}{16} ln ∣ x + 1 ∣ - \frac{1}{16 ( x + 1 )} - \frac{3}{16} ln ∣ x - 1 ∣ - \frac{1}{16 ( x - 1 )} : \frac{3}{8} arctan (x) + \frac{x}{8 ( 1 + x ^{2} )} + \frac{3}{16} ln ∣ x + 1 ∣ - \frac{1}{16 ( x + 1 )} - \frac{3}{16} ln ∣ x - 1 ∣ - \frac{1}{16 ( x - 1 )}

= \frac{3}{8} arctan (x) + \frac{x}{8 ( 1 + x ^{2} )} + \frac{3}{16} ln ∣ x + 1 ∣ - \frac{1}{16 ( x + 1 )} - \frac{3}{16} ln ∣ x - 1 ∣ - \frac{1}{16 ( x - 1 )}

定数を解答に追加する

= \frac{3}{8} arctan (x) + \frac{x}{8 ( 1 + x ^{2} )} + \frac{3}{16} ln ∣ x + 1 ∣ - \frac{1}{16 ( x + 1 )} - \frac{3}{16} ln ∣ x - 1 ∣ - \frac{1}{16 ( x - 1 )} + C