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

The integral from 0 to 1 of x(sqrt(x)-x^2) is 3/20

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

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

\frac{3}{20}

Decimal Notation

0.15

Solution steps

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

Expand

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

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

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

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

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

Simplify

\frac{2}{5} - \frac{1}{4} : \frac{3}{20}

= \frac{3}{20}

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What is the integral from 0 to 1 of x(sqrt(x)-x^2) ?
The integral from 0 to 1 of x(sqrt(x)-x^2) is 3/20