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受欢迎的微积分 >

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

解答

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

解答

\frac{1}{4} ln 4 x^{4} - 4 x^{2} + 8 + \frac{1}{2 7} arctan (\frac{2 x ^{2} - 1}{7}) + C

求解步骤

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

使用换元积分法

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

提出常数:

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

= \frac{1}{2} \cdot \int \frac{u}{u ^{2} + 2 - u} d u

对

u^{2} + 2 - u

配方:

(u - \frac{1}{2})^{2} + \frac{7}{4}

= \frac{1}{2} \cdot \int \frac{u}{( u - \frac{1}{2} ) ^{2} + \frac{7}{4}} d u

使用换元积分法

= \frac{1}{2} \cdot \int \frac{2 ( 2 v + 1 )}{4 v ^{2} + 7} d v

提出常数:

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

= \frac{1}{2} \cdot 2 \cdot \int \frac{2 v + 1}{4 v ^{2} + 7} d v

乘开

\frac{2 v + 1}{4 v ^{2} + 7} : \frac{2 v}{4 v ^{2} + 7} + \frac{1}{4 v ^{2} + 7}

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

= \frac{1}{2} \cdot 2 (\int \frac{2 v}{4 v ^{2} + 7} d v + \int \frac{1}{4 v ^{2} + 7} d v)

\int \frac{2 v}{4 v ^{2} + 7} d v = \frac{1}{4} ln 4 v^{2} + 7

\int \frac{1}{4 v ^{2} + 7} d v = \frac{1}{2 7} arctan (\frac{2}{7} v)

= \frac{1}{2} \cdot 2 (\frac{1}{4} ln 4 v^{2} + 7 + \frac{1}{2 7} arctan (\frac{2}{7} v))

代回

= \frac{1}{2} \cdot 2 (\frac{1}{4} ln 4 (x^{2} - \frac{1}{2})^{2} + 7 + \frac{1}{2 7} arctan (\frac{2}{7} (x^{2} - \frac{1}{2})))

化简

\frac{1}{2} \cdot 2 (\frac{1}{4} ln 4 (x^{2} - \frac{1}{2})^{2} + 7 + \frac{1}{2 7} arctan (\frac{2}{7} (x^{2} - \frac{1}{2}))) : \frac{1}{4} ln 4 x^{4} - 4 x^{2} + 8 + \frac{1}{2 7} arctan (\frac{2 x ^{2} - 1}{7})

= \frac{1}{4} ln 4 x^{4} - 4 x^{2} + 8 + \frac{1}{2 7} arctan (\frac{2 x ^{2} - 1}{7})

解答补常数

= \frac{1}{4} ln 4 x^{4} - 4 x^{2} + 8 + \frac{1}{2 7} arctan (\frac{2 x ^{2} - 1}{7}) + C

作图

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