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

解答

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

\frac{2}{3} x + \frac{5}{9} ln ∣ 3 x + 2 ∣ + C

求解步骤

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

长除

\frac{2 x + 3}{3 x + 2} : \frac{2}{3} + \frac{5}{3 ( 3 x + 2 )}

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

\int \frac{2}{3} d x = \frac{2}{3} x

\int \frac{5}{3 ( 3 x + 2 )} d x = \frac{5}{9} ln ∣ 3 x + 2 ∣

= \frac{2}{3} x + \frac{5}{9} ln ∣ 3 x + 2 ∣

解答补常数

= \frac{2}{3} x + \frac{5}{9} ln ∣ 3 x + 2 ∣ + C

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