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

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

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integral of 1/(x^{-1/2)-x}

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

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

Solution steps

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

Factor

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

= \int \frac{1}{\frac{1 - x ^{\frac{3}{2}}}{x ^{\frac{1}{2}}}} d x

Simplify

= \int \frac{x ^{\frac{1}{2}}}{1 - x ^{\frac{3}{2}}} d x

Apply u-substitution

= \int - \frac{2}{3 u} d u

Take the constant out:

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

= - \frac{2}{3} \cdot \int \frac{1}{u} d u

Use the common integral:

\int \frac{1}{u} d u = ln (∣ u ∣)

= - \frac{2}{3} ln ∣ u ∣

Substitute back

u = 1 - x^{\frac{3}{2}}

= - \frac{2}{3} ln 1 - x^{\frac{3}{2}}

Add a constant to the solution

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

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