What is the integral from 0 to 1 of x(1-sqrt(x))^2 ?

The integral from 0 to 1 of x(1-sqrt(x))^2 is 1/30

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integral from 0 to 1 of x(1-sqrt(x))^2

\int_{0}^{1} x (1 - x)^{2} d x

\frac{1}{30}

Decimal

0.03333 \dots

Solution steps

\int_{0}^{1} x (1 - x)^{2} d x

Expand

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

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

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

\int_{0}^{1} 2 x^{\frac{3}{2}} d x = \frac{4}{5}

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

= \frac{1}{2} - \frac{4}{5} + \frac{1}{3}

Simplify

\frac{1}{2} - \frac{4}{5} + \frac{1}{3} : \frac{1}{30}

= \frac{1}{30}

What is the integral from 0 to 1 of x(1-sqrt(x))^2 ?
The integral from 0 to 1 of x(1-sqrt(x))^2 is 1/30