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

The integral of ((x-x^2)^3)/(xsqrt(x)) is -2/11 x^{11/2}+2/3 x^{9/2}-6/7 x^{7/2}+2/5 x^{5/2}+C

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

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

- \frac{2}{11} x^{\frac{11}{2}} + \frac{2}{3} x^{\frac{9}{2}} - \frac{6}{7} x^{\frac{7}{2}} + \frac{2}{5} x^{\frac{5}{2}} + C

Solution steps

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

Expand

\frac{( x - x ^{2} ) ^{3}}{x x} : - x^{\frac{9}{2}} + 3 x^{\frac{7}{2}} - 3 x^{\frac{5}{2}} + x^{\frac{3}{2}}

= \int - x^{\frac{9}{2}} + 3 x^{\frac{7}{2}} - 3 x^{\frac{5}{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 x^{\frac{9}{2}} d x + \int 3 x^{\frac{7}{2}} d x - \int 3 x^{\frac{5}{2}} d x + \int x^{\frac{3}{2}} d x

\int x^{\frac{9}{2}} d x = \frac{2}{11} x^{\frac{11}{2}}

\int 3 x^{\frac{7}{2}} d x = \frac{2}{3} x^{\frac{9}{2}}

\int 3 x^{\frac{5}{2}} d x = \frac{6}{7} x^{\frac{7}{2}}

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

= - \frac{2}{11} x^{\frac{11}{2}} + \frac{2}{3} x^{\frac{9}{2}} - \frac{6}{7} x^{\frac{7}{2}} + \frac{2}{5} x^{\frac{5}{2}}

Add a constant to the solution

= - \frac{2}{11} x^{\frac{11}{2}} + \frac{2}{3} x^{\frac{9}{2}} - \frac{6}{7} x^{\frac{7}{2}} + \frac{2}{5} x^{\frac{5}{2}} + C

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What is the integral of ((x-x^2)^3)/(xsqrt(x)) ?
The integral of ((x-x^2)^3)/(xsqrt(x)) is -2/11 x^{11/2}+2/3 x^{9/2}-6/7 x^{7/2}+2/5 x^{5/2}+C