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

解答

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

- x + \frac{1}{2} (x + 2)^{2} - 2 + C

求解步骤

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

乘开

\frac{x ^{2} + 3 x + 2}{x + 2} : \frac{x ^{2}}{x + 2} + \frac{3 x}{x + 2} + \frac{2}{x + 2}

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

\int \frac{x ^{2}}{x + 2} d x = \frac{( x + 2 ) ^{2}}{2} - 4 (x + 2) + 4 ln ∣ x + 2 ∣

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

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

= \frac{( x + 2 ) ^{2}}{2} - 4 (x + 2) + 4 ln ∣ x + 2 ∣ + 3 (x + 2 - 2 ln ∣ x + 2 ∣) + 2 ln ∣ x + 2 ∣

化简

\frac{( x + 2 ) ^{2}}{2} - 4 (x + 2) + 4 ln ∣ x + 2 ∣ + 3 (x + 2 - 2 ln ∣ x + 2 ∣) + 2 ln ∣ x + 2 ∣ : - x + \frac{1}{2} (x + 2)^{2} - 2

= - x + \frac{1}{2} (x + 2)^{2} - 2

解答补常数

= - x + \frac{1}{2} (x + 2)^{2} - 2 + C