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

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

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integral of (u-2u^2)/(sqrt(u))

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

\frac{2}{3} u^{\frac{3}{2}} - \frac{4}{5} u^{\frac{5}{2}} + C

Solution steps

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

Expand

\frac{u - 2 u ^{2}}{u} : \frac{u}{u} - \frac{2 u ^{2}}{u}

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

\int \frac{u}{u} d u = \frac{2}{3} u^{\frac{3}{2}}

\int \frac{2 u ^{2}}{u} d u = \frac{4}{5} u^{\frac{5}{2}}

= \frac{2}{3} u^{\frac{3}{2}} - \frac{4}{5} u^{\frac{5}{2}}

Add a constant to the solution

= \frac{2}{3} u^{\frac{3}{2}} - \frac{4}{5} u^{\frac{5}{2}} + C

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