What is the integral from 5 to 13 of (5-x)(x-13) ?

The integral from 5 to 13 of (5-x)(x-13) is 256/3

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integral from 5 to 13 of (5-x)(x-13)

\int_{5}^{13} (5 - x) (x - 13) d x

\frac{256}{3}

Decimal

85.33333 \dots

Solution steps

\int_{5}^{13} (5 - x) (x - 13) d x

Expand

(5 - x) (x - 13) : 18 x - x^{2} - 65

= \int_{5}^{13} 18 x - x^{2} - 65 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_{5}^{13} 18 x d x - \int_{5}^{13} x^{2} d x - \int_{5}^{13} 65 d x

\int_{5}^{13} 18 x d x = 1296

\int_{5}^{13} x^{2} d x = \frac{2072}{3}

\int_{5}^{13} 65 d x = 520

= 1296 - \frac{2072}{3} - 520

Simplify

1296 - \frac{2072}{3} - 520 : \frac{256}{3}

= \frac{256}{3}

What is the integral from 5 to 13 of (5-x)(x-13) ?
The integral from 5 to 13 of (5-x)(x-13) is 256/3