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

The integral of ((1-x)^2)/(xsqrt(x)) is -2/(sqrt(x))-4sqrt(x)+2/3 x^{3/2}+C

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

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

- \frac{2}{x} - 4 x + \frac{2}{3} x^{\frac{3}{2}} + C

Solution steps

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

Simplify

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

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

Expand

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

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

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

\int \frac{2}{x ^{\frac{1}{2}}} d x = 4 x^{\frac{1}{2}}

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

= - \frac{2}{x} - 4 x^{\frac{1}{2}} + \frac{2}{3} x^{\frac{3}{2}}

Apply radical rule:

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

= - \frac{2}{x} - 4 x + \frac{2}{3} x^{\frac{3}{2}}

Add a constant to the solution

= - \frac{2}{x} - 4 x + \frac{2}{3} x^{\frac{3}{2}} + C

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What is the integral of ((1-x)^2)/(xsqrt(x)) ?
The integral of ((1-x)^2)/(xsqrt(x)) is -2/(sqrt(x))-4sqrt(x)+2/3 x^{3/2}+C