What is the integral from 0 to 1 of x^2(6x-6x^2) ?

The integral from 0 to 1 of x^2(6x-6x^2) is 3/10

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integral from 0 to 1 of x^2(6x-6x^2)

\int_{0}^{1} x^{2} (6 x - 6 x^{2}) d x

\frac{3}{10}

Decimal

0.3

Solution steps

\int_{0}^{1} x^{2} (6 x - 6 x^{2}) d x

Expand

x^{2} (6 x - 6 x^{2}) : 6 x^{3} - 6 x^{4}

= \int_{0}^{1} 6 x^{3} - 6 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

= \int_{0}^{1} 6 x^{3} d x - \int_{0}^{1} 6 x^{4} d x

\int_{0}^{1} 6 x^{3} d x = \frac{3}{2}

\int_{0}^{1} 6 x^{4} d x = \frac{6}{5}

= \frac{3}{2} - \frac{6}{5}

Simplify

\frac{3}{2} - \frac{6}{5} : \frac{3}{10}

= \frac{3}{10}

What is the integral from 0 to 1 of x^2(6x-6x^2) ?
The integral from 0 to 1 of x^2(6x-6x^2) is 3/10