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

The integral from 0 to x of 2(1-x) is 2x-x^2

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integral from 0 to x of 2(1-x)

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

2 x - x^{2}

Solution steps

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

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

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

= 2 (x - \frac{x ^{2}}{2})

Simplify

2 (x - \frac{x ^{2}}{2}) : 2 x - x^{2}

= 2 x - x^{2}

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