What is the integral from 0 to 2 of x(3/4 x(2-x)) ?

The integral from 0 to 2 of x(3/4 x(2-x)) is 1

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integral from 0 to 2 of x(3/4 x(2-x))

\int_{0}^{2} x (\frac{3}{4} x (2 - x)) d x

1

Solution steps

\int_{0}^{2} x \frac{3}{4} x (2 - x) d x

Take the constant out:

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

= \frac{3}{4} \cdot \int_{0}^{2} xx (2 - x) d x

Expand

xx (2 - x) : 2 x^{2} - x^{3}

= \frac{3}{4} \cdot \int_{0}^{2} 2 x^{2} - x^{3} 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

= \frac{3}{4} (\int_{0}^{2} 2 x^{2} d x - \int_{0}^{2} x^{3} d x)

\int_{0}^{2} 2 x^{2} d x = \frac{16}{3}

\int_{0}^{2} x^{3} d x = 4

= \frac{3}{4} (\frac{16}{3} - 4)

Simplify

\frac{3}{4} (\frac{16}{3} - 4) : 1

= 1

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What is the integral from 0 to 2 of x(3/4 x(2-x)) ?
The integral from 0 to 2 of x(3/4 x(2-x)) is 1