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

The integral of 1/(sqrt(1-x)-1) is -2(sqrt(1-x)-1+ln|sqrt(1-x)-1|)+C

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

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

- 2 (1 - x - 1 + ln 1 - x - 1) + C

Solution steps

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

Apply u-substitution

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

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= - 2 \cdot \int \frac{u}{u - 1} d u

Apply u-substitution

= - 2 \cdot \int \frac{v + 1}{v} d v

Expand

\frac{v + 1}{v} : 1 + \frac{1}{v}

= - 2 \cdot \int 1 + \frac{1}{v} d v

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

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

\int 1 d v = v

\int \frac{1}{v} d v = ln ∣ v ∣

= - 2 (v + ln ∣ v ∣)

Substitute back

= - 2 (1 - x - 1 + ln 1 - x - 1)

Add a constant to the solution

= - 2 (1 - x - 1 + ln 1 - x - 1) + C

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