integral of 1/((x-1)^2*(x^2+3))

解答

\int \frac{1}{( x - 1 ) ^{2} \cdot ( x ^{2} + 3 )} d x

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

求解步骤

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

将

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

用部份分式展开:

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

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

\int \frac{1}{8 ( x - 1 )} d x = \frac{1}{8} ln ∣ x - 1 ∣

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

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

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

解答补常数

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

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