What is the integral from 0 to 2 of x(sqrt(8x)-2x) ?

The integral from 0 to 2 of x(sqrt(8x)-2x) is 16/15

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integral from 0 to 2 of x(sqrt(8x)-2x)

\int_{0}^{2} x (8 x - 2 x) d x

\frac{16}{15}

Decimal

1.06666 \dots

Solution steps

\int_{0}^{2} x (8 x - 2 x) d x

Expand

x (8 x - 2 x) : 8 x^{\frac{3}{2}} - 2 x^{2}

= \int_{0}^{2} 8 x^{\frac{3}{2}} - 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}^{2} 8 x^{\frac{3}{2}} d x - \int_{0}^{2} 2 x^{2} d x

\int_{0}^{2} 8 x^{\frac{3}{2}} d x = \frac{32}{5}

\int_{0}^{2} 2 x^{2} d x = \frac{16}{3}

= \frac{32}{5} - \frac{16}{3}

Simplify

\frac{32}{5} - \frac{16}{3} : \frac{16}{15}

= \frac{16}{15}

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What is the integral from 0 to 2 of x(sqrt(8x)-2x) ?
The integral from 0 to 2 of x(sqrt(8x)-2x) is 16/15