What is the integral of ((2-x)^3)/(xsqrt(x)) ?

The integral of ((2-x)^3)/(xsqrt(x)) is -(16)/(sqrt(x))-24sqrt(x)+4x^{3/2}-2/5 x^{5/2}+C

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integral of ((2-x)^3)/(xsqrt(x))

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

- \frac{16}{x} - 24 x + 4 x^{\frac{3}{2}} - \frac{2}{5} x^{\frac{5}{2}} + C

Solution steps

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

Expand

\frac{( 2 - x ) ^{3}}{x x} : \frac{8}{x ^{\frac{3}{2}}} - \frac{12}{x ^{\frac{1}{2}}} + 6 x^{\frac{1}{2}} - x^{\frac{3}{2}}

= \int \frac{8}{x ^{\frac{3}{2}}} - \frac{12}{x ^{\frac{1}{2}}} + 6 x^{\frac{1}{2}} - x^{\frac{3}{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{8}{x ^{\frac{3}{2}}} d x - \int \frac{12}{x ^{\frac{1}{2}}} d x + \int 6 x^{\frac{1}{2}} d x - \int x^{\frac{3}{2}} d x

\int \frac{8}{x ^{\frac{3}{2}}} d x = - \frac{16}{x}

\int \frac{12}{x ^{\frac{1}{2}}} d x = 24 x^{\frac{1}{2}}

\int 6 x^{\frac{1}{2}} d x = 4 x^{\frac{3}{2}}

\int x^{\frac{3}{2}} d x = \frac{2}{5} x^{\frac{5}{2}}

= - \frac{16}{x} - 24 x^{\frac{1}{2}} + 4 x^{\frac{3}{2}} - \frac{2}{5} x^{\frac{5}{2}}

Apply radical rule:

= - \frac{16}{x} - 24 x + 4 x^{\frac{3}{2}} - \frac{2}{5} x^{\frac{5}{2}}

Add a constant to the solution

= - \frac{16}{x} - 24 x + 4 x^{\frac{3}{2}} - \frac{2}{5} x^{\frac{5}{2}} + C

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What is the integral of ((2-x)^3)/(xsqrt(x)) ?
The integral of ((2-x)^3)/(xsqrt(x)) is -(16)/(sqrt(x))-24sqrt(x)+4x^{3/2}-2/5 x^{5/2}+C