What is the integral from 0 to 12 of 2pix(12x-x^2) ?

The integral from 0 to 12 of 2pix(12x-x^2) is 3456pi

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integral from 0 to 12 of 2pix(12x-x^2)

\int_{0}^{12} 2 π x (12 x - x^{2}) d x

3456 π

Decimal Notation

10857.34421 \dots

Solution steps

\int_{0}^{12} 2 π x (12 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}^{12} x (12 x - x^{2}) d x

Expand

x (12 x - x^{2}) : 12 x^{2} - x^{3}

= 2 π \cdot \int_{0}^{12} 12 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}^{12} 12 x^{2} d x - \int_{0}^{12} x^{3} d x)

\int_{0}^{12} 12 x^{2} d x = 6912

\int_{0}^{12} x^{3} d x = 5184

= 2 π (6912 - 5184)

Simplify

= 3456 π

(u - u) (u + u)

x^{2} + 2 x - 3

x^{2} y^{^{'}} = 2 x^{2} e^{4 x} - 3 x y

\frac{d}{d t} (\frac{cos ( t ) + sin ( t )}{- sin ( t ) + cos ( t )})

\int \frac{26}{x + x x} d x

What is the integral from 0 to 12 of 2pix(12x-x^2) ?
The integral from 0 to 12 of 2pix(12x-x^2) is 3456pi