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

解答

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

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

求解步骤

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

分解

\frac{x ^{3}}{x ^{2} + x} : \frac{x ^{2}}{x + 1}

= \int \frac{x ^{2}}{x + 1} d x

使用换元积分法

= \int \frac{( u - 1 ) ^{2}}{u} d u

乘开

\frac{( u - 1 ) ^{2}}{u} : u - 2 + \frac{1}{u}

= \int u - 2 + \frac{1}{u} 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 u d u - \int 2 d u + \int \frac{1}{u} d u

\int u d u = \frac{u ^{2}}{2}

\int 2 d u = 2 u

\int \frac{1}{u} d u = ln ∣ u ∣

= \frac{u ^{2}}{2} - 2 u + ln ∣ u ∣

u = x + 1

代回

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

解答补常数

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