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受欢迎的微积分 >

integral of ((4x-5))/((x-2)(x+3))

解答

\int \frac{( 4 x - 5 )}{( x - 2 ) ( x + 3 )} d x

解答

\frac{17}{5} ln ∣ x + 3 ∣ + \frac{3}{5} ln ∣ x - 2 ∣ + C

求解步骤

\int \frac{4 x - 5}{( x - 2 ) ( x + 3 )} d x

使用换元积分法

= \int \frac{4 u - 17}{u ( u - 5 )} d u

将

\frac{4 u - 17}{u ( u - 5 )}

用部份分式展开:

\frac{17}{5 u} + \frac{3}{5 ( u - 5 )}

= \int \frac{17}{5 u} + \frac{3}{5 ( u - 5 )} 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

= \int \frac{17}{5 u} d u + \int \frac{3}{5 ( u - 5 )} d u

\int \frac{17}{5 u} d u = \frac{17}{5} ln ∣ u ∣

\int \frac{3}{5 ( u - 5 )} d u = \frac{3}{5} ln ∣ u - 5 ∣

= \frac{17}{5} ln ∣ u ∣ + \frac{3}{5} ln ∣ u - 5 ∣

u = x + 3

代回

= \frac{17}{5} ln ∣ x + 3 ∣ + \frac{3}{5} ln ∣ x + 3 - 5 ∣

化简

\frac{17}{5} ln ∣ x + 3 ∣ + \frac{3}{5} ln ∣ x + 3 - 5 ∣ : \frac{17}{5} ln ∣ x + 3 ∣ + \frac{3}{5} ln ∣ x - 2 ∣

= \frac{17}{5} ln ∣ x + 3 ∣ + \frac{3}{5} ln ∣ x - 2 ∣

解答补常数

= \frac{17}{5} ln ∣ x + 3 ∣ + \frac{3}{5} ln ∣ x - 2 ∣ + C

作图

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