integral of (x^4+x^3+6x+3)/(x^3-x^2)

解答

\int \frac{x ^{4} + x ^{3} + 6 x + 3}{x ^{3} - x ^{2}} d x

2 x + \frac{1}{2} (x - 1)^{2} + 11 ln ∣ x - 1 ∣ + \frac{x ^{2}}{x - 1} - \frac{1}{x - 1} - 9 ln ∣ x ∣ + \frac{3}{x} - 2 + C

求解步骤

\int \frac{x ^{4} + x ^{3} + 6 x + 3}{x ^{3} - x ^{2}} d x

乘开

\frac{x ^{4} + x ^{3} + 6 x + 3}{x ^{3} - x ^{2}} : \frac{x ^{4}}{x ^{3} - x ^{2}} + \frac{x ^{3}}{x ^{3} - x ^{2}} + \frac{6 x}{x ^{3} - x ^{2}} + \frac{3}{x ^{3} - 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 ^{4}}{x ^{3} - x ^{2}} d x + \int \frac{x ^{3}}{x ^{3} - x ^{2}} d x + \int \frac{6 x}{x ^{3} - x ^{2}} d x + \int \frac{3}{x ^{3} - x ^{2}} d x

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

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

\int \frac{6 x}{x ^{3} - x ^{2}} d x = 6 (- ln ∣ x ∣ + ln ∣ x - 1 ∣)

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

= \frac{( x - 1 ) ^{2}}{2} + 2 (x - 1) + ln ∣ x - 1 ∣ + \frac{x ^{2}}{x - 1} + ln ∣ x - 1 ∣ - \frac{1}{x - 1} + 6 (- ln ∣ x ∣ + ln ∣ x - 1 ∣) + 3 (- ln ∣ x ∣ + \frac{1}{x} + ln ∣ x - 1 ∣)

化简

\frac{( x - 1 ) ^{2}}{2} + 2 (x - 1) + ln ∣ x - 1 ∣ + \frac{x ^{2}}{x - 1} + ln ∣ x - 1 ∣ - \frac{1}{x - 1} + 6 (- ln ∣ x ∣ + ln ∣ x - 1 ∣) + 3 (- ln ∣ x ∣ + \frac{1}{x} + ln ∣ x - 1 ∣) : 2 x + \frac{1}{2} (x - 1)^{2} + 11 ln ∣ x - 1 ∣ + \frac{x ^{2}}{x - 1} - \frac{1}{x - 1} - 9 ln ∣ x ∣ + \frac{3}{x} - 2

= 2 x + \frac{1}{2} (x - 1)^{2} + 11 ln ∣ x - 1 ∣ + \frac{x ^{2}}{x - 1} - \frac{1}{x - 1} - 9 ln ∣ x ∣ + \frac{3}{x} - 2

解答补常数

= 2 x + \frac{1}{2} (x - 1)^{2} + 11 ln ∣ x - 1 ∣ + \frac{x ^{2}}{x - 1} - \frac{1}{x - 1} - 9 ln ∣ x ∣ + \frac{3}{x} - 2 + C

\int \frac{e ^{- x}}{e ^{x} + e ^{- x}} d x

\int e x^{3} d x

\int 4 x^{2} - 3 x d x

\int (u^{3} - u e^{6 - 3 u^{2}}) d u

\int x^{4} (x^{5} - 1)^{3} d x