What is the integral from 0 to 1 of 2(ax-(x^2+x)) ?

The integral from 0 to 1 of 2(ax-(x^2+x)) is a-5/3

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integral from 0 to 1 of 2(ax-(x^2+x))

\int_{0}^{1} 2 (a x - (x^{2} + x)) d x

a - \frac{5}{3}

Solution steps

\int_{0}^{1} 2 (a x - (x^{2} + x)) d x

Take the constant out:

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

= 2 \cdot \int_{0}^{1} a x - (x^{2} + x) 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

= 2 (\int_{0}^{1} a x d x - \int_{0}^{1} x^{2} + x d x)

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

\int_{0}^{1} x^{2} + x d x = \frac{5}{6}

= 2 (\frac{a}{2} - \frac{5}{6})

Simplify

2 (\frac{a}{2} - \frac{5}{6}) : a - \frac{5}{3}

= a - \frac{5}{3}

What is the integral from 0 to 1 of 2(ax-(x^2+x)) ?
The integral from 0 to 1 of 2(ax-(x^2+x)) is a-5/3