What is the integral from 0.5 to 1 of 3/4 x(1-0.25x) ?

The integral from 0.5 to 1 of 3/4 x(1-0.25x) is 0.2265625

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integral from 0.5 to 1 of 3/4 x(1-0.25x)

\int_{0.5}^{1} \frac{3}{4} x (1 - 0.25 x) d x

0.2265625

Solution steps

\int_{0.5}^{1} \frac{3}{4} x (1 - 0.25 x) d x

Take the constant out:

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

= \frac{3}{4} \cdot \int_{0.5}^{1} x (1 - 0.25 x) d x

Expand

x (1 - 0.25 x) : x - 0.25 x^{2}

= \frac{3}{4} \cdot \int_{0.5}^{1} x - 0.25 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

= \frac{3}{4} (\int_{0.5}^{1} x d x - \int_{0.5}^{1} 0.25 x^{2} d x)

\int_{0.5}^{1} x d x = 0.375

\int_{0.5}^{1} 0.25 x^{2} d x = 0.07291 \dots

= \frac{3}{4} (0.375 - 0.07291 \dots)

Simplify

\frac{3}{4} (0.375 - 0.07291 \dots) : 0.2265625

= 0.2265625

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What is the integral from 0.5 to 1 of 3/4 x(1-0.25x) ?
The integral from 0.5 to 1 of 3/4 x(1-0.25x) is 0.2265625