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

解答

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

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

求解步骤

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

将

\frac{1}{( x - 1 ) ( x ^{2} - 1 ) ^{2}}

用部份分式展开:

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

= \int \frac{3}{16 ( x - 1 )} - \frac{1}{4 ( x - 1 ) ^{2}} + \frac{1}{4 ( x - 1 ) ^{3}} - \frac{3}{16 ( x + 1 )} - \frac{1}{8 ( x + 1 ) ^{2}} d x

使用积分加法定则:

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

= \int \frac{3}{16 ( x - 1 )} d x - \int \frac{1}{4 ( x - 1 ) ^{2}} d x + \int \frac{1}{4 ( x - 1 ) ^{3}} d x - \int \frac{3}{16 ( x + 1 )} d x - \int \frac{1}{8 ( x + 1 ) ^{2}} d x

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

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

\int \frac{1}{4 ( x - 1 ) ^{3}} d x = - \frac{1}{8 ( x - 1 ) ^{2}}

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

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

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

化简

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

解答补常数

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

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