What is the integral from 0 to 0.4 of 30x^2(1-x^2) ?

The integral from 0 to 0.4 of 30x^2(1-x^2) is 0.57856

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integral from 0 to 0.4 of 30x^2(1-x^2)

\int_{0}^{0.4} 30 x^{2} (1 - x^{2}) d x

0.57856

Solution steps

\int_{0}^{0.4} 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}^{0.4} x^{2} (1 - x^{2}) d x

Expand

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

= 30 \cdot \int_{0}^{0.4} x^{2} - 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}^{0.4} x^{2} d x - \int_{0}^{0.4} x^{4} d x)

\int_{0}^{0.4} x^{2} d x = 0.02133 \dots

\int_{0}^{0.4} x^{4} d x = 0.002048

= 30 (0.02133 \dots - 0.002048)

Simplify

= 0.57856

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What is the integral from 0 to 0.4 of 30x^2(1-x^2) ?
The integral from 0 to 0.4 of 30x^2(1-x^2) is 0.57856