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

The integral from 1 to 2 of 2x(x-1) is 5/3

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

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

\frac{5}{3}

Decimal Notation

1.66666 \dots

Solution steps

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

Take the constant out:

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

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

Expand

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

= 2 \cdot \int_{1}^{2} 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_{1}^{2} x^{2} d x - \int_{1}^{2} x d x)

\int_{1}^{2} x^{2} d x = \frac{7}{3}

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

= 2 (\frac{7}{3} - \frac{3}{2})

Simplify

2 (\frac{7}{3} - \frac{3}{2}) : \frac{5}{3}

= \frac{5}{3}

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What is the integral from 1 to 2 of 2x(x-1) ?
The integral from 1 to 2 of 2x(x-1) is 5/3