What is the integral from 0 to α of x*α/(x^{α+1)} ?

The integral from 0 to α of x*α/(x^{α+1)} is (α^{-α+2})/(-α+1)

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integral from 0 to α of x*α/(x^{α+1)}

\int_{0}^{α} x \cdot \frac{α}{x ^{α + 1}} d x

\frac{α ^{- α + 2}}{- α + 1}

Solution steps

\int_{0}^{α} x \frac{α}{x ^{α + 1}} d x

Take the constant out:

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

= α \cdot \int_{0}^{α} x \frac{1}{x ^{α + 1}} d x

Simplify

x \frac{1}{x ^{α + 1}} : \frac{1}{x ^{α}}

= α \cdot \int_{0}^{α} \frac{1}{x ^{α}} d x

Apply the Power Rule

: [\frac{x ^{- α + 1}}{- α + 1}]_{0}^{α}

= α [\frac{x ^{- α + 1}}{- α + 1}]_{0}^{α}

Compute the boundaries:

\frac{α ^{- α + 1}}{- α + 1}

= α \frac{α ^{- α + 1}}{- α + 1}

Simplify

= \frac{α ^{- α + 2}}{- α + 1}

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What is the integral from 0 to α of x*α/(x^{α+1)} ?
The integral from 0 to α of x*α/(x^{α+1)} is (α^{-α+2})/(-α+1)