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

The integral from 0 to 2 of 2pix(2x-x^2) is (8pi)/3

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

\int_{0}^{2} 2 π x (2 x - x^{2}) d x

\frac{8 π}{3}

Decimal

8.37758 \dots

Solution steps

\int_{0}^{2} 2 π x (2 x - x^{2}) d x

Take the constant out:

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

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

Expand

x (2 x - x^{2}) : 2 x^{2} - x^{3}

= 2 π \cdot \int_{0}^{2} 2 x^{2} - x^{3} 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

= 2 π (\int_{0}^{2} 2 x^{2} d x - \int_{0}^{2} x^{3} d x)

\int_{0}^{2} 2 x^{2} d x = \frac{16}{3}

\int_{0}^{2} x^{3} d x = 4

= 2 π (\frac{16}{3} - 4)

Simplify

2 π (\frac{16}{3} - 4) : \frac{8 π}{3}

= \frac{8 π}{3}

\int cot (18 x) d x

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\frac{1}{( s ( s ^{2} + 1 ) )}

\frac{d}{d s} (s^{3})

n = 0 \sum \infty \frac{5 ^{n}}{6 ^{n}}

What is the integral from 0 to 2 of 2pix(2x-x^2) ?
The integral from 0 to 2 of 2pix(2x-x^2) is (8pi)/3