integral of (x^3-x+2)/(x-1)

解答

\int \frac{x ^{3} - x + 2}{x - 1} d x

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

求解步骤

\int \frac{x ^{3} - x + 2}{x - 1} d x

乘开

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

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 ^{3}}{x - 1} d x - \int \frac{x}{x - 1} d x + \int \frac{2}{x - 1} d x

\int \frac{x ^{3}}{x - 1} d x = \frac{( x - 1 ) ^{3}}{3} + \frac{3 ( x - 1 ) ^{2}}{2} + 3 (x - 1) + ln ∣ x - 1 ∣

\int \frac{x}{x - 1} d x = x - 1 + ln ∣ x - 1 ∣

\int \frac{2}{x - 1} d x = 2 ln ∣ x - 1 ∣

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

化简

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

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

解答补常数

= 2 x + 2 ln ∣ x - 1 ∣ + \frac{1}{3} (x - 1)^{3} + \frac{3}{2} (x - 1)^{2} - 2 + C

l a pl a ce t r an s f or m e^{- 2 t} cos^{2} (t)

\frac{\partial}{\partial y} (24 y^{2} - 12 x)

\frac{\partial}{\partial x} (7 Y (x))

in v erse l a pl a ce \frac{- 5}{s ( 6 ( 5 s + 1 ) )}

d er i v a t i v e f (x) = (7 x^{6} + 4 x^{3})^{4}