What is the integral from 0 to 1 of 20(x^3-x^4) ?

The integral from 0 to 1 of 20(x^3-x^4) is 1

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integral from 0 to 1 of 20(x^3-x^4)

\int_{0}^{1} 20 (x^{3} - x^{4}) d x

1

Solution steps

\int_{0}^{1} 20 (x^{3} - x^{4}) d x

Take the constant out:

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

= 20 \cdot \int_{0}^{1} 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

= 20 (\int_{0}^{1} x^{3} d x - \int_{0}^{1} x^{4} d x)

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

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

= 20 (\frac{1}{4} - \frac{1}{5})

Simplify

20 (\frac{1}{4} - \frac{1}{5}) : 1

= 1

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What is the integral from 0 to 1 of 20(x^3-x^4) ?
The integral from 0 to 1 of 20(x^3-x^4) is 1