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

The integral from 0 to x of 20x^3(1-x) is 20((x^4)/4-(x^5)/5)

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

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

20 (\frac{x ^{4}}{4} - \frac{x ^{5}}{5})

Solution steps

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

Take the constant out:

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

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

Expand

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

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

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

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

= 20 (\frac{x ^{4}}{4} - \frac{x ^{5}}{5})

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What is the integral from 0 to x of 20x^3(1-x) ?
The integral from 0 to x of 20x^3(1-x) is 20((x^4)/4-(x^5)/5)