What is the integral of (x+sqrt(x))(4-x) ?

The integral of (x+sqrt(x))(4-x) is 2x^2-(x^3)/3+8/3 x^{3/2}-2/5 x^{5/2}+C

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integral of (x+sqrt(x))(4-x)

\int (x + x) (4 - x) d x

2 x^{2} - \frac{x ^{3}}{3} + \frac{8}{3} x^{\frac{3}{2}} - \frac{2}{5} x^{\frac{5}{2}} + C

Solution steps

\int (x + x) (4 - x) d x

Expand

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

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

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

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

\int 4 x d x = \frac{8}{3} x^{\frac{3}{2}}

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

= 2 x^{2} - \frac{x ^{3}}{3} + \frac{8}{3} x^{\frac{3}{2}} - \frac{2}{5} x^{\frac{5}{2}}

Add a constant to the solution

= 2 x^{2} - \frac{x ^{3}}{3} + \frac{8}{3} x^{\frac{3}{2}} - \frac{2}{5} x^{\frac{5}{2}} + C

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What is the integral of (x+sqrt(x))(4-x) ?
The integral of (x+sqrt(x))(4-x) is 2x^2-(x^3)/3+8/3 x^{3/2}-2/5 x^{5/2}+C