What is the integral from 0 to 1 of 1/2 x+1/2-x^2 ?

The integral from 0 to 1 of 1/2 x+1/2-x^2 is 5/12

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integral from 0 to 1 of 1/2 x+1/2-x^2

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

\frac{5}{12}

Decimal

0.41666 \dots

Solution steps

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

\int_{0}^{1} \frac{1}{2} x d x = \frac{1}{4}

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

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

= \frac{1}{4} + \frac{1}{2} - \frac{1}{3}

Simplify

\frac{1}{4} + \frac{1}{2} - \frac{1}{3} : \frac{5}{12}

= \frac{5}{12}

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What is the integral from 0 to 1 of 1/2 x+1/2-x^2 ?
The integral from 0 to 1 of 1/2 x+1/2-x^2 is 5/12