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受欢迎的微积分 >

integral of (x^3+2x^2-18x+2)/(x^2+x-12)

解答

\int \frac{x ^{3} + 2 x ^{2} - 18 x + 2}{x ^{2} + x - 12} d x

解答

2 (\frac{1}{16} (2 x + 1)^{2} + \frac{1}{4} (x + \frac{1}{2}) - \frac{7}{4} ln 4 x^{2} + 4 x - 48 - \frac{5}{4} (ln \frac{8}{7} + \frac{2}{7} x - ln - \frac{6}{7} + \frac{2}{7} x)) + C

求解步骤

\int \frac{x ^{3} + 2 x ^{2} - 18 x + 2}{x ^{2} + x - 12} d x

对

x^{2} + x - 12

配方:

(x + \frac{1}{2})^{2} - \frac{49}{4}

= \int \frac{x ^{3} + 2 x ^{2} - 18 x + 2}{( x + \frac{1}{2} ) ^{2} - \frac{49}{4}} d x

使用换元积分法

= \int \frac{2 ( 4 u ^{2} - 40 u + 2 ( u - \frac{1}{2} ) ^{3} + 23 )}{4 u ^{2} - 49} 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 ^{2} - 40 u + 2 ( u - \frac{1}{2} ) ^{3} + 23}{4 u ^{2} - 49} d u

长除

\frac{4 u ^{2} - 40 u + 2 ( u - \frac{1}{2} ) ^{3} + 23}{4 u ^{2} - 49} : \frac{u}{2} + \frac{1}{4} + \frac{- 14 u + 35}{4 u ^{2} - 49}

= 2 \cdot \int \frac{u}{2} + \frac{1}{4} + \frac{- 14 u + 35}{4 u ^{2} - 49} d u

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{u}{2} d u + \int \frac{1}{4} d u + \int \frac{- 14 u + 35}{4 u ^{2} - 49} d u)

\int \frac{u}{2} d u = \frac{u ^{2}}{4}

\int \frac{1}{4} d u = \frac{1}{4} u

\int \frac{- 14 u + 35}{4 u ^{2} - 49} d u = - \frac{7}{4} ln 4 u^{2} - 49 - \frac{5}{4} (ln \frac{2}{7} u + 1 - ln \frac{2}{7} u - 1)

= 2 (\frac{u ^{2}}{4} + \frac{1}{4} u - \frac{7}{4} ln 4 u^{2} - 49 - \frac{5}{4} (ln \frac{2}{7} u + 1 - ln \frac{2}{7} u - 1))

u = x + \frac{1}{2}

代回

= 2 (\frac{( x + \frac{1}{2} ) ^{2}}{4} + \frac{1}{4} (x + \frac{1}{2}) - \frac{7}{4} ln 4 (x + \frac{1}{2})^{2} - 49 - \frac{5}{4} (ln \frac{2}{7} (x + \frac{1}{2}) + 1 - ln \frac{2}{7} (x + \frac{1}{2}) - 1))

化简

2 (\frac{( x + \frac{1}{2} ) ^{2}}{4} + \frac{1}{4} (x + \frac{1}{2}) - \frac{7}{4} ln 4 (x + \frac{1}{2})^{2} - 49 - \frac{5}{4} (ln \frac{2}{7} (x + \frac{1}{2}) + 1 - ln \frac{2}{7} (x + \frac{1}{2}) - 1)) : 2 (\frac{1}{16} (2 x + 1)^{2} + \frac{1}{4} (x + \frac{1}{2}) - \frac{7}{4} ln 4 x^{2} + 4 x - 48 - \frac{5}{4} (ln \frac{8}{7} + \frac{2}{7} x - ln - \frac{6}{7} + \frac{2}{7} x))

= 2 (\frac{1}{16} (2 x + 1)^{2} + \frac{1}{4} (x + \frac{1}{2}) - \frac{7}{4} ln 4 x^{2} + 4 x - 48 - \frac{5}{4} (ln \frac{8}{7} + \frac{2}{7} x - ln - \frac{6}{7} + \frac{2}{7} x))

解答补常数

= 2 (\frac{1}{16} (2 x + 1)^{2} + \frac{1}{4} (x + \frac{1}{2}) - \frac{7}{4} ln 4 x^{2} + 4 x - 48 - \frac{5}{4} (ln \frac{8}{7} + \frac{2}{7} x - ln - \frac{6}{7} + \frac{2}{7} x)) + C

作图

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