What is the integral of 1/(x^2)sqrt(1/x-1) ?

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

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integral of 1/(x^2)sqrt(1/x-1)

\int \frac{1}{x ^{2}} \frac{1}{x} - 1 d x

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

Solution steps

\int \frac{1}{x ^{2}} \frac{1}{x} - 1 d x

Apply u-substitution

= \int - u - 1 d u

Take the constant out:

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

= - \int u - 1 d u

Apply u-substitution

= - \int v d v

Apply the Power Rule

= - \frac{2}{3} v^{\frac{3}{2}}

Substitute back

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

Join

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

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

Add a constant to the solution

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

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