What is the integral from 1 to 4 of sqrt(x)(2-x) ?

The integral from 1 to 4 of sqrt(x)(2-x) is -46/15

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integral from 1 to 4 of sqrt(x)(2-x)

\int_{1}^{4} x (2 - x) d x

- \frac{46}{15}

Decimal

- 3.06666 \dots

Solution steps

\int_{1}^{4} x (2 - x) d x

Expand

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

= \int_{1}^{4} 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_{1}^{4} 2 x d x - \int_{1}^{4} x^{\frac{3}{2}} d x

\int_{1}^{4} 2 x d x = \frac{28}{3}

\int_{1}^{4} x^{\frac{3}{2}} d x = \frac{62}{5}

= \frac{28}{3} - \frac{62}{5}

Simplify

\frac{28}{3} - \frac{62}{5} : - \frac{46}{15}

= - \frac{46}{15}

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What is the integral from 1 to 4 of sqrt(x)(2-x) ?
The integral from 1 to 4 of sqrt(x)(2-x) is -46/15