What is the integral from 0 to 2 of x*1/2 (2-x) ?

The integral from 0 to 2 of x*1/2 (2-x) is 2/3

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integral from 0 to 2 of x*1/2 (2-x)

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

\frac{2}{3}

Decimal Notation

0.66666 \dots

Solution steps

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

Expand

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

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

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

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

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

Simplify

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

= \frac{2}{3}

\frac{d}{d x} (6 x^{3} e^{- x})

f (x) = 3 x ln (x)

\frac{1}{3} x^{3} - 7 x + 5

n = 1 \sum \infty \frac{x ^{n}}{2 ^{n}}

x \to 0 lim (\frac{f ( x )}{g ( x )})

What is the integral from 0 to 2 of x*1/2 (2-x) ?
The integral from 0 to 2 of x*1/2 (2-x) is 2/3