What is the integral from 0 to 1 of x(x-\sqrt[5]{x}) ?

The integral from 0 to 1 of x(x-\sqrt[5]{x}) is -4/33

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integral from 0 to 1 of x(x-\sqrt[5]{x})

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

- \frac{4}{33}

Decimal

- 0.12121 \dots

Solution steps

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

Expand

x (x - 5 x) : x^{2} - x^{\frac{6}{5}}

= \int_{0}^{1} x^{2} - x^{\frac{6}{5}} 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^{2} d x - \int_{0}^{1} x^{\frac{6}{5}} d x

\int_{0}^{1} x^{2} d x = \frac{1}{3}

\int_{0}^{1} x^{\frac{6}{5}} d x = \frac{5}{11}

= \frac{1}{3} - \frac{5}{11}

Simplify

\frac{1}{3} - \frac{5}{11} : - \frac{4}{33}

= - \frac{4}{33}

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What is the integral from 0 to 1 of x(x-\sqrt[5]{x}) ?
The integral from 0 to 1 of x(x-\sqrt[5]{x}) is -4/33