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

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

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integral from 0 to 1 of 5(1-x)^4

\int_{0}^{1} 5 (1 - x)^{4} d x

1

Solution steps

\int_{0}^{1} 5 (1 - x)^{4} d x

Take the constant out:

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

= 5 \cdot \int_{0}^{1} (1 - x)^{4} d x

Apply u-substitution

= 5 \cdot \int_{1}^{0} - u^{4} d u

\int_{a}^{b} f (x) d x = - \int_{b}^{a} f (x) d x, a < b

= 5 (- \int_{0}^{1} - u^{4} d u)

Take the constant out:

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

= 5 (- (- \int_{0}^{1} u^{4} d u))

Apply the Power Rule

= 5 (- (- [\frac{u ^{5}}{5}]_{0}^{1}))

Simplify

= 5 [\frac{u ^{5}}{5}]_{0}^{1}

Compute the boundaries:

\frac{1}{5}

= 5 \cdot \frac{1}{5}

Simplify

= 1

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