What is the integral from 0 to 1 of xθx^{θ-1} ?

The integral from 0 to 1 of xθx^{θ-1} is θ/(θ+1)

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integral from 0 to 1 of xθx^{θ-1}

\int_{0}^{1} x θ x^{θ - 1} d x

\frac{θ}{θ + 1}

Solution steps

\int_{0}^{1} x θ x^{θ - 1} d x

Take the constant out:

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

= θ \cdot \int_{0}^{1} x x^{- 1 + θ} d x

Simplify

x x^{- 1 + θ} : x^{θ}

= θ \cdot \int_{0}^{1} x^{θ} d x

Apply the Power Rule

: [\frac{x ^{θ + 1}}{θ + 1}]_{0}^{1}

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

Compute the boundaries:

\frac{1}{θ + 1}

= θ \frac{1}{θ + 1}

Simplify

= \frac{θ}{θ + 1}

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What is the integral from 0 to 1 of xθx^{θ-1} ?
The integral from 0 to 1 of xθx^{θ-1} is θ/(θ+1)