What is the integral of x(x-sqrt(x)) ?

The integral of x(x-sqrt(x)) is (x^3)/2-2/3 x^{5/2}-(x^3)/6+4/15 x^{5/2}+C

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integral of x(x-sqrt(x))

\int x (x - x) d x

\frac{x ^{3}}{2} - \frac{2}{3} x^{\frac{5}{2}} - \frac{x ^{3}}{6} + \frac{4}{15} x^{\frac{5}{2}} + C

Solution steps

\int x (x - x) d x

Apply Integration By Parts

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

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

\int \frac{x ^{2}}{2} - \frac{2}{3} x^{\frac{3}{2}} d x = \frac{x ^{3}}{6} - \frac{4}{15} x^{\frac{5}{2}}

= x (\frac{x ^{2}}{2} - \frac{2}{3} x^{\frac{3}{2}}) - (\frac{x ^{3}}{6} - \frac{4}{15} x^{\frac{5}{2}})

Simplify

x (\frac{x ^{2}}{2} - \frac{2}{3} x^{\frac{3}{2}}) - (\frac{x ^{3}}{6} - \frac{4}{15} x^{\frac{5}{2}}) : \frac{x ^{3}}{2} - \frac{2}{3} x^{\frac{5}{2}} - \frac{x ^{3}}{6} + \frac{4}{15} x^{\frac{5}{2}}

= \frac{x ^{3}}{2} - \frac{2}{3} x^{\frac{5}{2}} - \frac{x ^{3}}{6} + \frac{4}{15} x^{\frac{5}{2}}

Add a constant to the solution

= \frac{x ^{3}}{2} - \frac{2}{3} x^{\frac{5}{2}} - \frac{x ^{3}}{6} + \frac{4}{15} x^{\frac{5}{2}} + C

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What is the integral of x(x-sqrt(x)) ?
The integral of x(x-sqrt(x)) is (x^3)/2-2/3 x^{5/2}-(x^3)/6+4/15 x^{5/2}+C