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

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

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

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

\frac{1}{15}

Decimal

0.06666 \dots

Solution steps

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

Expand

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

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

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

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

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

Simplify

\frac{2}{5} - \frac{1}{3} : \frac{1}{15}

= \frac{1}{15}

\frac{d}{d x} (ln (x - 2))

\int_{0}^{π} (5 e^{x} + 5 sin (x)) d x

d er i v a t i v e f (x) = \frac{1}{2 x}

\frac{\partial}{\partial x} (sin (x y^{2}))

d er i v a t i v e \frac{x ^{2} - 2}{e ^{2 x}}

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