What is the integral from 0 to 1 of sqrt(x)(1-x) ?

The integral from 0 to 1 of sqrt(x)(1-x) is 4/15

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integral from 0 to 1 of sqrt(x)(1-x)

\int_{0}^{1} x (1 - x) d x

\frac{4}{15}

Decimal

0.26666 \dots

Solution steps

\int_{0}^{1} x (1 - x) d x

Expand

x (1 - x) : x - x^{\frac{3}{2}}

= \int_{0}^{1} 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_{0}^{1} x d x - \int_{0}^{1} x^{\frac{3}{2}} d x

\int_{0}^{1} x d x = \frac{2}{3}

\int_{0}^{1} x^{\frac{3}{2}} d x = \frac{2}{5}

= \frac{2}{3} - \frac{2}{5}

Simplify

\frac{2}{3} - \frac{2}{5} : \frac{4}{15}

= \frac{4}{15}

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What is the integral from 0 to 1 of sqrt(x)(1-x) ?
The integral from 0 to 1 of sqrt(x)(1-x) is 4/15