What is the integral from 0 to 1 of 2piy(y-y^7) ?

The integral from 0 to 1 of 2piy(y-y^7) is (4pi)/9

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integral from 0 to 1 of 2piy(y-y^7)

\int_{0}^{1} 2 π y (y - y^{7}) d y

\frac{4 π}{9}

Decimal Notation

1.39626 \dots

Solution steps

\int_{0}^{1} 2 π y (y - y^{7}) d y

Take the constant out:

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

= 2 π \cdot \int_{0}^{1} y (y - y^{7}) d y

Expand

y (y - y^{7}) : y^{2} - y^{8}

= 2 π \cdot \int_{0}^{1} y^{2} - y^{8} d y

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}^{1} y^{2} d y - \int_{0}^{1} y^{8} d y)

\int_{0}^{1} y^{2} d y = \frac{1}{3}

\int_{0}^{1} y^{8} d y = \frac{1}{9}

= 2 π (\frac{1}{3} - \frac{1}{9})

Simplify

2 π (\frac{1}{3} - \frac{1}{9}) : \frac{4 π}{9}

= \frac{4 π}{9}

\int - 2 x^{5} d x

\frac{\partial}{\partial y} (x + 2 - y)

\int_{0}^{5} \frac{1}{x ^{\frac{12}{13}}} d x

x^{2} - 4, 0, - 4, 3

\frac{\partial}{\partial y} (cos (x - y))

What is the integral from 0 to 1 of 2piy(y-y^7) ?
The integral from 0 to 1 of 2piy(y-y^7) is (4pi)/9