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

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

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

\int (\frac{x - x}{x ^{2}}) d x

- \frac{2}{x} - ln ∣ x ∣ + C

Solution steps

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

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

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

Simplify

\frac{x}{x ^{2}} - \frac{x}{x ^{2}} : \frac{1}{x ^{\frac{3}{2}}} - \frac{1}{x}

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

Apply the Sum Rule:

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

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

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

\int \frac{1}{x} d x = ln ∣ x ∣

= - \frac{2}{x} - ln ∣ x ∣

Add a constant to the solution

= - \frac{2}{x} - ln ∣ x ∣ + C

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