What is the integral from 0 to 2 of ((x-2)^2)/2 ?

The integral from 0 to 2 of ((x-2)^2)/2 is 4/3

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integral from 0 to 2 of ((x-2)^2)/2

\int_{0}^{2} \frac{( x - 2 ) ^{2}}{2} d x

\frac{4}{3}

Decimal

1.33333 \dots

Solution steps

\int_{0}^{2} \frac{( x - 2 ) ^{2}}{2} d x

Take the constant out:

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

= \frac{1}{2} \cdot \int_{0}^{2} (x - 2)^{2} d x

Expand

(x - 2)^{2} : x^{2} - 4 x + 4

= \frac{1}{2} \cdot \int_{0}^{2} x^{2} - 4 x + 4 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{1}{2} (\int_{0}^{2} x^{2} d x - \int_{0}^{2} 4 x d x + \int_{0}^{2} 4 d x)

\int_{0}^{2} x^{2} d x = \frac{8}{3}

\int_{0}^{2} 4 x d x = 8

\int_{0}^{2} 4 d x = 8

= \frac{1}{2} (\frac{8}{3} - 8 + 8)

Simplify

\frac{1}{2} (\frac{8}{3} - 8 + 8) : \frac{4}{3}

= \frac{4}{3}

What is the integral from 0 to 2 of ((x-2)^2)/2 ?
The integral from 0 to 2 of ((x-2)^2)/2 is 4/3