What is the integral from 0 to 1 of 30x^2(1-x)^2 ?

The integral from 0 to 1 of 30x^2(1-x)^2 is 1

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integral from 0 to 1 of 30x^2(1-x)^2

\int_{0}^{1} 30 x^{2} (1 - x)^{2} d x

1

Solution steps

\int_{0}^{1} 30 x^{2} (1 - x)^{2} d x

Take the constant out:

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

= 30 \cdot \int_{0}^{1} x^{2} (1 - x)^{2} d x

Expand

x^{2} (1 - x)^{2} : x^{2} - 2 x^{3} + x^{4}

= 30 \cdot \int_{0}^{1} x^{2} - 2 x^{3} + x^{4} 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

= 30 (\int_{0}^{1} x^{2} d x - \int_{0}^{1} 2 x^{3} d x + \int_{0}^{1} x^{4} d x)

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

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

\int_{0}^{1} x^{4} d x = \frac{1}{5}

= 30 (\frac{1}{3} - \frac{1}{2} + \frac{1}{5})

Simplify

30 (\frac{1}{3} - \frac{1}{2} + \frac{1}{5}) : 1

= 1

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What is the integral from 0 to 1 of 30x^2(1-x)^2 ?
The integral from 0 to 1 of 30x^2(1-x)^2 is 1