integral of (xsqrt(x)-1-x-sqrt(x))/(x-1)

解答

\int \frac{x x - 1 - x - x}{x - 1} d x

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

求解步骤

\int \frac{x x - 1 - x - x}{x - 1} d x

乘开

\frac{x x - 1 - x - x}{x - 1} : \frac{x x}{x - 1} - \frac{1}{x - 1} - \frac{x}{x - 1} - \frac{x}{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 x}{x - 1} d x - \int \frac{1}{x - 1} d x - \int \frac{x}{x - 1} d x - \int \frac{x}{x - 1} d x

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

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

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

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

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

化简

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

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

解答补常数

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

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