What is the integral from 3 to 5 of x(2-sqrt(x)) ?

The integral from 3 to 5 of x(2-sqrt(x)) is 16-10sqrt(5)+(18sqrt(3))/5

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integral from 3 to 5 of x(2-sqrt(x))

\int_{3}^{5} x (2 - x) d x

16 - 105 + \frac{18 3}{5}

Decimal

- 0.12529 \dots

Solution steps

\int_{3}^{5} x (2 - x) d x

Expand

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

= \int_{3}^{5} 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_{3}^{5} 2 x d x - \int_{3}^{5} x^{\frac{3}{2}} d x

\int_{3}^{5} 2 x d x = 16

\int_{3}^{5} x^{\frac{3}{2}} d x = 105 - \frac{18 3}{5}

= 16 - (105 - \frac{18 3}{5})

Simplify

= 16 - 105 + \frac{18 3}{5}

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What is the integral from 3 to 5 of x(2-sqrt(x)) ?
The integral from 3 to 5 of x(2-sqrt(x)) is 16-10sqrt(5)+(18sqrt(3))/5