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

The integral of sqrt(x/(x-1)) is sqrt(x/(x-1))(x-1)+ln|sqrt(x)+sqrt(x-1)|+C

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

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

\frac{x}{x - 1} (x - 1) + ln x + x - 1 + C

Solution steps

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

Apply u-substitution

= \int \frac{u + 1}{u} d u

Apply Integration By Parts

= \frac{u + 1}{u} u - \int - \frac{1}{2 u ^{\frac{1}{2}} u + 1} d u

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

= \frac{u + 1}{u} u - (- ln u + 1 + u)

Substitute back

u = x - 1

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

Simplify

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

Add a constant to the solution

= \frac{x}{x - 1} (x - 1) + ln x + x - 1 + C

What is the integral of sqrt(x/(x-1)) ?
The integral of sqrt(x/(x-1)) is sqrt(x/(x-1))(x-1)+ln|sqrt(x)+sqrt(x-1)|+C