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

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

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

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

- 2 ln 1 - x + C

Solution steps

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

Apply u-substitution

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

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

Take the constant out:

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

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

Apply u-substitution

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

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

Take the constant out:

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

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

Use the common integral:

\int \frac{1}{v} d v = ln (∣ v ∣)

= 2 (- ln ∣ v ∣)

Substitute back

= 2 (- ln 1 - x)

Simplify

= - 2 ln 1 - x

Add a constant to the solution

= - 2 ln 1 - x + C

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