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

The integral from 0 to 1 of 2pix(1-sqrt(x)) is pi/5

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

\int_{0}^{1} 2 π x (1 - x) d x

\frac{π}{5}

Decimal Notation

0.62831 \dots

Solution steps

\int_{0}^{1} 2 π x (1 - 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 (1 - x) d x

Expand

x (1 - x) : x - x^{\frac{3}{2}}

= 2 π \cdot \int_{0}^{1} x - x^{\frac{3}{2}} 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}^{1} x d x - \int_{0}^{1} x^{\frac{3}{2}} d x)

\int_{0}^{1} x d x = \frac{1}{2}

\int_{0}^{1} x^{\frac{3}{2}} d x = \frac{2}{5}

= 2 π (\frac{1}{2} - \frac{2}{5})

Simplify

2 π (\frac{1}{2} - \frac{2}{5}) : \frac{π}{5}

= \frac{π}{5}

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What is the integral from 0 to 1 of 2pix(1-sqrt(x)) ?
The integral from 0 to 1 of 2pix(1-sqrt(x)) is pi/5