integral of (3x)/((x+2)(x+3))

解答

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

3 (- 2 ln ∣ x + 2 ∣ + 3 ln ∣ x + 3 ∣) + C

求解步骤

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

提出常数:

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

= 3 \cdot \int \frac{x}{( x + 2 ) ( x + 3 )} d x

将

\frac{x}{( x + 2 ) ( x + 3 )}

用部份分式展开:

- \frac{2}{x + 2} + \frac{3}{x + 3}

= 3 \cdot \int - \frac{2}{x + 2} + \frac{3}{x + 3} 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

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

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

\int \frac{3}{x + 3} d x = 3 ln ∣ x + 3 ∣

= 3 (- 2 ln ∣ x + 2 ∣ + 3 ln ∣ x + 3 ∣)

解答补常数

= 3 (- 2 ln ∣ x + 2 ∣ + 3 ln ∣ x + 3 ∣) + C

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