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受欢迎的微积分 >

integral of (x+2)/(x^2+9x-10)

解答

\int \frac{x + 2}{x ^{2} + 9 x - 10} d x

解答

2 (\frac{1}{4} ln 4 x^{2} + 36 x - 40 + \frac{5}{44} (ln \frac{20}{11} + \frac{2}{11} x - ln - \frac{2}{11} + \frac{2}{11} x)) + C

求解步骤

\int \frac{x + 2}{x ^{2} + 9 x - 10} d x

对

x^{2} + 9 x - 10

配方:

(x + \frac{9}{2})^{2} - \frac{121}{4}

= \int \frac{x + 2}{( x + \frac{9}{2} ) ^{2} - \frac{121}{4}} d x

使用换元积分法

= \int \frac{2 ( 2 u - 5 )}{4 u ^{2} - 121} d u

提出常数:

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

= 2 \cdot \int \frac{2 u - 5}{4 u ^{2} - 121} d u

乘开

\frac{2 u - 5}{4 u ^{2} - 121} : \frac{2 u}{4 u ^{2} - 121} - \frac{5}{4 u ^{2} - 121}

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{2 u}{4 u ^{2} - 121} d u - \int \frac{5}{4 u ^{2} - 121} d u)

\int \frac{2 u}{4 u ^{2} - 121} d u = \frac{1}{4} ln 4 u^{2} - 121

\int \frac{5}{4 u ^{2} - 121} d u = - \frac{5}{44} (ln \frac{2}{11} u + 1 - ln \frac{2}{11} u - 1)

= 2 (\frac{1}{4} ln 4 u^{2} - 121 - (- \frac{5}{44} (ln \frac{2}{11} u + 1 - ln \frac{2}{11} u - 1)))

u = x + \frac{9}{2}

代回

= 2 (\frac{1}{4} ln 4 (x + \frac{9}{2})^{2} - 121 - (- \frac{5}{44} (ln \frac{2}{11} (x + \frac{9}{2}) + 1 - ln \frac{2}{11} (x + \frac{9}{2}) - 1)))

化简

2 (\frac{1}{4} ln 4 (x + \frac{9}{2})^{2} - 121 - (- \frac{5}{44} (ln \frac{2}{11} (x + \frac{9}{2}) + 1 - ln \frac{2}{11} (x + \frac{9}{2}) - 1))) : 2 (\frac{1}{4} ln 4 x^{2} + 36 x - 40 + \frac{5}{44} (ln \frac{20}{11} + \frac{2}{11} x - ln - \frac{2}{11} + \frac{2}{11} x))

= 2 (\frac{1}{4} ln 4 x^{2} + 36 x - 40 + \frac{5}{44} (ln \frac{20}{11} + \frac{2}{11} x - ln - \frac{2}{11} + \frac{2}{11} x))

解答补常数

= 2 (\frac{1}{4} ln 4 x^{2} + 36 x - 40 + \frac{5}{44} (ln \frac{20}{11} + \frac{2}{11} x - ln - \frac{2}{11} + \frac{2}{11} x)) + C

作图

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