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受欢迎的微积分 >

integral of (-1(x-9))/((x+5)(x-2))

解答

\int \frac{- 1 ( x - 9 )}{( x + 5 ) ( x - 2 )} d x

解答

- 2 ln ∣ x + 5 ∣ + ln ∣ x - 2 ∣ + C

求解步骤

\int \frac{- 1 \cdot ( x - 9 )}{( x + 5 ) ( x - 2 )} d x

提出常数:

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

= - \int \frac{x - 9}{( x + 5 ) ( x - 2 )} d x

将

\frac{x - 9}{( x + 5 ) ( x - 2 )}

用部份分式展开:

\frac{2}{x + 5} - \frac{1}{x - 2}

= - \int \frac{2}{x + 5} - \frac{1}{x - 2} d x

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

= - (\int \frac{2}{x + 5} d x - \int \frac{1}{x - 2} d x)

\int \frac{2}{x + 5} d x = 2 ln ∣ x + 5 ∣

\int \frac{1}{x - 2} d x = ln ∣ x - 2 ∣

= - (2 ln ∣ x + 5 ∣ - ln ∣ x - 2 ∣)

化简

= - 2 ln ∣ x + 5 ∣ + ln ∣ x - 2 ∣

解答补常数

= - 2 ln ∣ x + 5 ∣ + ln ∣ x - 2 ∣ + C

作图

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