What is the integral from 0 to 1 of 2pix(21-21x) ?

The integral from 0 to 1 of 2pix(21-21x) is 7pi

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integral from 0 to 1 of 2pix(21-21x)

\int_{0}^{1} 2 π x (21 - 21 x) d x

7 π

Decimal

21.99114 \dots

Solution steps

\int_{0}^{1} 2 π x (21 - 21 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 (21 - 21 x) d x

Apply Integration By Parts

= 2 π [\frac{1}{2} x^{2} (21 - 21 x) - \int - \frac{21 x ^{2}}{2} d x]_{0}^{1}

\int - \frac{21 x ^{2}}{2} d x = - \frac{7 x ^{3}}{2}

= 2 π [\frac{1}{2} x^{2} (21 - 21 x) - (- \frac{7 x ^{3}}{2})]_{0}^{1}

Simplify

2 π [\frac{1}{2} x^{2} (21 - 21 x) - (- \frac{7 x ^{3}}{2})]_{0}^{1} : 2 π [\frac{1}{2} (21 x^{2} - 14 x^{3})]_{0}^{1}

= 2 π [\frac{1}{2} (21 x^{2} - 14 x^{3})]_{0}^{1}

Compute the boundaries:

\frac{7}{2}

= 2 π \frac{7}{2}

Simplify

= 7 π

What is the integral from 0 to 1 of 2pix(21-21x) ?
The integral from 0 to 1 of 2pix(21-21x) is 7pi