What is the integral of (2-x)sqrt(x) ?

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

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integral of (2-x)sqrt(x)

\int (2 - x) x d x

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

Solution steps

\int (2 - x) x d x

Expand

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

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

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

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

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

Add a constant to the solution

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

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