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

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

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

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

- \frac{2}{x} - 2 x + C

Solution steps

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

Expand

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

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

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

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

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

Apply radical rule:

a^{\frac{1}{n}} = n a

= - \frac{2}{x} - 2 x

Add a constant to the solution

= - \frac{2}{x} - 2 x + C

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sin (x) 1 - cos (x)

\int \frac{3}{x ^{2} - 9} d x

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