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

解答

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

- \frac{1}{2 2} ln 2 x^{2} + 22 x + 2 + \frac{1}{2 2} ln 2 x^{2} - 22 x + 2 + C

求解步骤

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

乘开

\frac{x ^{2} - 1}{x ^{4} + 1} : \frac{x ^{2}}{x ^{4} + 1} - \frac{1}{x ^{4} + 1}

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

\int \frac{x ^{2}}{x ^{4} + 1} d x = - \frac{1}{4 2} (ln 2 x^{2} + 22 x + 2 - 2 arctan (2 x + 1)) + \frac{1}{4 2} (ln 2 x^{2} - 22 x + 2 + 2 arctan (2 x - 1))

\int \frac{1}{x ^{4} + 1} d x = \frac{1}{4 2} (ln 2 x^{2} + 22 x + 2 + 2 arctan (2 x + 1)) + \frac{1}{4 2} (- ln 2 x^{2} - 22 x + 2 + 2 arctan (2 x - 1))

= - \frac{1}{4 2} (ln 2 x^{2} + 22 x + 2 - 2 arctan (2 x + 1)) + \frac{1}{4 2} (ln 2 x^{2} - 22 x + 2 + 2 arctan (2 x - 1)) - (\frac{1}{4 2} (ln 2 x^{2} + 22 x + 2 + 2 arctan (2 x + 1)) + \frac{1}{4 2} (- ln 2 x^{2} - 22 x + 2 + 2 arctan (2 x - 1)))

化简

- \frac{1}{4 2} (ln 2 x^{2} + 22 x + 2 - 2 arctan (2 x + 1)) + \frac{1}{4 2} (ln 2 x^{2} - 22 x + 2 + 2 arctan (2 x - 1)) - (\frac{1}{4 2} (ln 2 x^{2} + 22 x + 2 + 2 arctan (2 x + 1)) + \frac{1}{4 2} (- ln 2 x^{2} - 22 x + 2 + 2 arctan (2 x - 1))) : - \frac{1}{2 2} ln 2 x^{2} + 22 x + 2 + \frac{1}{2 2} ln 2 x^{2} - 22 x + 2

= - \frac{1}{2 2} ln 2 x^{2} + 22 x + 2 + \frac{1}{2 2} ln 2 x^{2} - 22 x + 2

解答补常数

= - \frac{1}{2 2} ln 2 x^{2} + 22 x + 2 + \frac{1}{2 2} ln 2 x^{2} - 22 x + 2 + C

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