integral of (3x^4-12)/(x^2+2)

解答

\int \frac{3 x ^{4} - 12}{x ^{2} + 2} d x

3 (\frac{x ^{3}}{2} - \frac{x ^{3}}{6} - 2 x) + C

求解步骤

\int \frac{3 x ^{4} - 12}{x ^{2} + 2} d x

分解

\frac{3 x ^{4} - 12}{x ^{2} + 2} : 3 (x + 2) (x - 2)

= \int 3 (x + 2) (x - 2) d x

提出常数:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= 3 \cdot \int (x + 2) (x - 2) d x

使用分布积分法

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

\int \frac{x ^{2}}{2} - 2 x d x = \frac{x ^{3}}{6} - \frac{x ^{2}}{2}

= 3 ((x + 2) (\frac{x ^{2}}{2} - 2 x) - (\frac{x ^{3}}{6} - \frac{x ^{2}}{2}))

乘开

(x + 2) (\frac{x ^{2}}{2} - 2 x) - (\frac{x ^{3}}{6} - \frac{x ^{2}}{2}) : \frac{x ^{3}}{2} - \frac{x ^{3}}{6} - 2 x

= 3 (\frac{x ^{3}}{2} - \frac{x ^{3}}{6} - 2 x)

解答补常数

= 3 (\frac{x ^{3}}{2} - \frac{x ^{3}}{6} - 2 x) + C

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