What is the integral of (xsqrt(x)-1-x-sqrt(x))/(x-1) ?

The integral of (xsqrt(x)-1-x-sqrt(x))/(x-1) is -x+2/3 x^{3/2}-2ln|x-1|+1+C

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

\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

Solution steps

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

Expand

\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}

Apply the Sum Rule:

\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)

Simplify

\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

Add a constant to the solution

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

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What is the integral of (xsqrt(x)-1-x-sqrt(x))/(x-1) ?
The integral of (xsqrt(x)-1-x-sqrt(x))/(x-1) is -x+2/3 x^{3/2}-2ln|x-1|+1+C