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

解答

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

- \frac{3}{2} arctan (\frac{x}{2}) + x + C

求解步骤

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

\frac{x ^{2} - 1}{x ^{2} + 2} = - \frac{3}{x ^{2} + 2} + 1

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

\int \frac{3}{x ^{2} + 2} d x = \frac{3}{2} arctan (\frac{x}{2})

\int 1 d x = x

= - \frac{3}{2} arctan (\frac{x}{2}) + x

解答补常数

= - \frac{3}{2} arctan (\frac{x}{2}) + x + C

\int \frac{e ^{- 2 x}}{- 2} d x

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