What is the integral from 0 to 1 of 2pix(6-6x) ?

The integral from 0 to 1 of 2pix(6-6x) is 2pi

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integral from 0 to 1 of 2pix(6-6x)

\int_{0}^{1} 2 π x (6 - 6 x) d x

2 π

Solution steps

\int_{0}^{1} 2 π x (6 - 6 x) 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}^{1} x (6 - 6 x) d x

Apply Integration By Parts

: [3 x^{2} (- x + 1) - \int - 3 x^{2} d x]_{0}^{1}

= 2 π [3 x^{2} (- x + 1) - \int - 3 x^{2} d x]_{0}^{1}

\int - 3 x^{2} d x = - x^{3}

= 2 π [3 x^{2} (- x + 1) - (- x^{3})]_{0}^{1}

Simplify

2 π [3 x^{2} (- x + 1) - (- x^{3})]_{0}^{1} : 2 π [- 2 x^{3} + 3 x^{2}]_{0}^{1}

= 2 π [- 2 x^{3} + 3 x^{2}]_{0}^{1}

Compute the boundaries:

1

= 2 π 1

Simplify

= 2 π

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What is the integral from 0 to 1 of 2pix(6-6x) ?
The integral from 0 to 1 of 2pix(6-6x) is 2pi