What is the integral from 0 to 15 of 2pix(15x-x^2) ?

The integral from 0 to 15 of 2pix(15x-x^2) is (16875pi)/2

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integral from 0 to 15 of 2pix(15x-x^2)

\int_{0}^{15} 2 π x (15 x - x^{2}) d x

\frac{16875 π}{2}

Solution steps

\int_{0}^{15} 2 π x (15 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}^{15} x (15 x - x^{2}) d x

Expand

x (15 x - x^{2}) : 15 x^{2} - x^{3}

= 2 π \cdot \int_{0}^{15} 15 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}^{15} 15 x^{2} d x - \int_{0}^{15} x^{3} d x)

\int_{0}^{15} 15 x^{2} d x = 16875

\int_{0}^{15} x^{3} d x = \frac{50625}{4}

= 2 π (16875 - \frac{50625}{4})

Simplify

2 π (16875 - \frac{50625}{4}) : \frac{16875 π}{2}

= \frac{16875 π}{2}

\frac{\partial}{\partial x} (ln (\frac{2 x y}{5 z}))

\int - \frac{9}{x ^{10}} d x

\frac{\partial}{\partial x} (u)

\frac{d}{d x} (4 π x)

x \to - 2 lim (\frac{1}{2 x})

What is the integral from 0 to 15 of 2pix(15x-x^2) ?
The integral from 0 to 15 of 2pix(15x-x^2) is (16875pi)/2