What is the integral from 2 to 7 of (2-x)(x-7) ?

The integral from 2 to 7 of (2-x)(x-7) is 125/6

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integral from 2 to 7 of (2-x)(x-7)

\int_{2}^{7} (2 - x) (x - 7) d x

\frac{125}{6}

Decimal

20.83333 \dots

Solution steps

\int_{2}^{7} (2 - x) (x - 7) d x

Expand

(2 - x) (x - 7) : 9 x - x^{2} - 14

= \int_{2}^{7} 9 x - x^{2} - 14 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_{2}^{7} 9 x d x - \int_{2}^{7} x^{2} d x - \int_{2}^{7} 14 d x

\int_{2}^{7} 9 x d x = \frac{405}{2}

\int_{2}^{7} x^{2} d x = \frac{335}{3}

\int_{2}^{7} 14 d x = 70

= \frac{405}{2} - \frac{335}{3} - 70

Simplify

\frac{405}{2} - \frac{335}{3} - 70 : \frac{125}{6}

= \frac{125}{6}

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What is the integral from 2 to 7 of (2-x)(x-7) ?
The integral from 2 to 7 of (2-x)(x-7) is 125/6