What is the integral from 0 to 1 of 2pix(12-12x) ?

The integral from 0 to 1 of 2pix(12-12x) is 4pi

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integral from 0 to 1 of 2pix(12-12x)

\int_{0}^{1} 2 π x (12 - 12 x) d x

4 π

Decimal Notation

12.56637 \dots

Solution steps

\int_{0}^{1} 2 π x (12 - 12 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 (12 - 12 x) d x

Apply Integration By Parts

: [6 x^{2} (- x + 1) - \int - 6 x^{2} d x]_{0}^{1}

= 2 π [6 x^{2} (- x + 1) - \int - 6 x^{2} d x]_{0}^{1}

\int - 6 x^{2} d x = - 2 x^{3}

= 2 π [6 x^{2} (- x + 1) - (- 2 x^{3})]_{0}^{1}

Simplify

2 π [6 x^{2} (- x + 1) - (- 2 x^{3})]_{0}^{1} : 2 π [- 4 x^{3} + 6 x^{2}]_{0}^{1}

= 2 π [- 4 x^{3} + 6 x^{2}]_{0}^{1}

Compute the boundaries:

2

= 2 π 2

Simplify

= 4 π

e^{4}

y^{^{''}} - \frac{9}{t} y^{^{'}} + \frac{34}{t ^{2}} y = 0

\frac{\partial}{\partial x} (\frac{1}{2 1 + x + y})

\frac{d y}{d x} = \frac{3 x + x y ^{2}}{y + x ^{2} y}

\int \frac{e ^{- x}}{1 - e ^{- 2 x}} d x

What is the integral from 0 to 1 of 2pix(12-12x) ?
The integral from 0 to 1 of 2pix(12-12x) is 4pi