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

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

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

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

0.16666 \dots

Solution steps

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

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

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

Expand

x (0.5 - x) : 0.5 x - x^{2}

= 8 \cdot \int_{0}^{0.5} 0.5 x - 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

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

\int_{0}^{0.5} 0.5 x d x = 0.0625

\int_{0}^{0.5} x^{2} d x = 0.04166 \dots

= 8 (0.0625 - 0.04166 \dots)

Simplify

= 0.16666 \dots

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\frac{d y}{d x} = \frac{2}{x ^{2}}

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x \to 0 lim (\frac{1}{9 - 3 ^{\frac{1}{x}}})

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