What is the integral from y/n to 1 of n(1-x)^{n-1} ?

The integral from y/n to 1 of n(1-x)^{n-1} is ((n-y)^n)/(n^n)

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integral from y/n to 1 of n(1-x)^{n-1}

\int_{\frac{y}{n}}^{1} n (1 - x)^{n - 1} d x

\frac{( n - y ) ^{n}}{n ^{n}}

Solution steps

\int_{\frac{y}{n}}^{1} n (1 - x)^{n - 1} d x

Take the constant out:

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

= n \cdot \int_{\frac{y}{n}}^{1} (1 - x)^{- 1 + n} d x

Apply u-substitution

: \int_{1 - \frac{y}{n}}^{0} - u^{- 1 + n} d u

= n \cdot \int_{1 - \frac{y}{n}}^{0} - u^{- 1 + n} d u

Take the constant out:

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

= n (- \int_{1 - \frac{y}{n}}^{0} u^{- 1 + n} d u)

Apply the Power Rule

: [\frac{u ^{n}}{n}]_{1 - \frac{y}{n}}^{0}

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

Simplify

= - n [\frac{u ^{n}}{n}]_{1 - \frac{y}{n}}^{0}

Compute the boundaries:

- \frac{( n - y ) ^{n}}{n ^{n + 1}}

= - n (- \frac{( n - y ) ^{n}}{n ^{n + 1}})

Simplify

= \frac{( n - y ) ^{n}}{n ^{n}}

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What is the integral from y/n to 1 of n(1-x)^{n-1} ?
The integral from y/n to 1 of n(1-x)^{n-1} is ((n-y)^n)/(n^n)