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

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

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

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

- 2 1 - x + C

Solution steps

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

Simplify

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

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

Take the constant out:

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

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

Simplify

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

Apply u-substitution

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

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

Take the constant out:

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

= 1 \cdot (- \int \frac{1}{u} d u)

Apply the Power Rule

: 2 u^{\frac{1}{2}}

= 1 \cdot (- 2 u^{\frac{1}{2}})

Substitute back

u = 1 - x

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

Simplify

1 \cdot (- 2 (1 - x)^{\frac{1}{2}}) : - 2 1 - x

= - 2 1 - x

Add a constant to the solution

= - 2 1 - x + C

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