What is the integral from 1 to 5 of (1-x)(x-5) ?

The integral from 1 to 5 of (1-x)(x-5) is 32/3

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integral from 1 to 5 of (1-x)(x-5)

\int_{1}^{5} (1 - x) (x - 5) d x

\frac{32}{3}

Decimal Notation

10.66666 \dots

Solution steps

\int_{1}^{5} (1 - x) (x - 5) d x

Expand

(1 - x) (x - 5) : 6 x - x^{2} - 5

= \int_{1}^{5} 6 x - x^{2} - 5 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_{1}^{5} 6 x d x - \int_{1}^{5} x^{2} d x - \int_{1}^{5} 5 d x

\int_{1}^{5} 6 x d x = 72

\int_{1}^{5} x^{2} d x = \frac{124}{3}

\int_{1}^{5} 5 d x = 20

= 72 - \frac{124}{3} - 20

Simplify

72 - \frac{124}{3} - 20 : \frac{32}{3}

= \frac{32}{3}

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What is the integral from 1 to 5 of (1-x)(x-5) ?
The integral from 1 to 5 of (1-x)(x-5) is 32/3