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

The integral of (x-1)/(sqrt(x)-1) is 3x-2sqrt(x)+2/3 (sqrt(x)-1)^3-1+C

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

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

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

Solution steps

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

Expand

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

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

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

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

Simplify

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

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

Add a constant to the solution

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

What is the integral of (x-1)/(sqrt(x)-1) ?
The integral of (x-1)/(sqrt(x)-1) is 3x-2sqrt(x)+2/3 (sqrt(x)-1)^3-1+C