What is the integral from 0 to 1 of x-x^k ?

The integral from 0 to 1 of x-x^k is 1/2-1/(k+1)

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integral from 0 to 1 of x-x^k

\int_{0}^{1} x - x^{k} d x

\frac{1}{2} - \frac{1}{k + 1}

Solution steps

\int_{0}^{1} x - x^{k} 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

= \int_{0}^{1} x d x - \int_{0}^{1} x^{k} d x

\int_{0}^{1} x d x = \frac{1}{2}

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

= \frac{1}{2} - \frac{1}{k + 1}

What is the integral from 0 to 1 of x-x^k ?
The integral from 0 to 1 of x-x^k is 1/2-1/(k+1)