What is the integral from 1 to 3 of 3/20 (x-3)^2 ?

The integral from 1 to 3 of 3/20 (x-3)^2 is 2/5

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integral from 1 to 3 of 3/20 (x-3)^2

\int_{1}^{3} \frac{3}{20} (x - 3)^{2} d x

\frac{2}{5}

Decimal

0.4

Solution steps

\int_{1}^{3} \frac{3}{20} (x - 3)^{2} d x

Take the constant out:

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

= \frac{3}{20} \cdot \int_{1}^{3} (x - 3)^{2} d x

Expand

(x - 3)^{2} : x^{2} - 6 x + 9

= \frac{3}{20} \cdot \int_{1}^{3} x^{2} - 6 x + 9 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

= \frac{3}{20} (\int_{1}^{3} x^{2} d x - \int_{1}^{3} 6 x d x + \int_{1}^{3} 9 d x)

\int_{1}^{3} x^{2} d x = \frac{26}{3}

\int_{1}^{3} 6 x d x = 24

\int_{1}^{3} 9 d x = 18

= \frac{3}{20} (\frac{26}{3} - 24 + 18)

Simplify

\frac{3}{20} (\frac{26}{3} - 24 + 18) : \frac{2}{5}

= \frac{2}{5}

What is the integral from 1 to 3 of 3/20 (x-3)^2 ?
The integral from 1 to 3 of 3/20 (x-3)^2 is 2/5