What is the integral from 0 to 0.5 of x(4-8x) ?

The integral from 0 to 0.5 of x(4-8x) is 0.16666…

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integral from 0 to 0.5 of x(4-8x)

\int_{0}^{0.5} x (4 - 8 x) d x

0.16666 \dots

Solution steps

\int_{0}^{0.5} x (4 - 8 x) d x

Expand

x (4 - 8 x) : 4 x - 8 x^{2}

= \int_{0}^{0.5} 4 x - 8 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}^{0.5} 4 x d x - \int_{0}^{0.5} 8 x^{2} d x

\int_{0}^{0.5} 4 x d x = 0.5

\int_{0}^{0.5} 8 x^{2} d x = 0.33333 \dots

= 0.5 - 0.33333 \dots

Simplify

= 0.16666 \dots

What is the integral from 0 to 0.5 of x(4-8x) ?
The integral from 0 to 0.5 of x(4-8x) is 0.16666…