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受欢迎的微积分 >

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

解答

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

解答

2 (\frac{1}{8} ln 16 x^{2} - 8 x + 8 + \frac{1}{4 7} arctan (\frac{4 x - 1}{7})) + C

求解步骤

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

对

2 x^{2} - x + 1

配方:

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

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

使用换元积分法

= \int \frac{2 ( 4 u + 1 )}{16 u ^{2} + 7} d u

提出常数:

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

= 2 \cdot \int \frac{4 u + 1}{16 u ^{2} + 7} d u

乘开

\frac{4 u + 1}{16 u ^{2} + 7} : \frac{4 u}{16 u ^{2} + 7} + \frac{1}{16 u ^{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

= 2 (\int \frac{4 u}{16 u ^{2} + 7} d u + \int \frac{1}{16 u ^{2} + 7} d u)

\int \frac{4 u}{16 u ^{2} + 7} d u = \frac{1}{8} ln 16 u^{2} + 7

\int \frac{1}{16 u ^{2} + 7} d u = \frac{1}{4 7} arctan (\frac{4}{7} u)

= 2 (\frac{1}{8} ln 16 u^{2} + 7 + \frac{1}{4 7} arctan (\frac{4}{7} u))

u = x - \frac{1}{4}

代回

= 2 (\frac{1}{8} ln 16 (x - \frac{1}{4})^{2} + 7 + \frac{1}{4 7} arctan (\frac{4}{7} (x - \frac{1}{4})))

化简

2 (\frac{1}{8} ln 16 (x - \frac{1}{4})^{2} + 7 + \frac{1}{4 7} arctan (\frac{4}{7} (x - \frac{1}{4}))) : 2 (\frac{1}{8} ln 16 x^{2} - 8 x + 8 + \frac{1}{4 7} arctan (\frac{4 x - 1}{7}))

= 2 (\frac{1}{8} ln 16 x^{2} - 8 x + 8 + \frac{1}{4 7} arctan (\frac{4 x - 1}{7}))

解答补常数

= 2 (\frac{1}{8} ln 16 x^{2} - 8 x + 8 + \frac{1}{4 7} arctan (\frac{4 x - 1}{7})) + C

作图

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