What is the integral from 0 to 1 of (x-sqrt(x))/3 ?

The integral from 0 to 1 of (x-sqrt(x))/3 is -1/18

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integral from 0 to 1 of (x-sqrt(x))/3

\int_{0}^{1} \frac{x - x}{3} d x

- \frac{1}{18}

Decimal Notation

- 0.05555 \dots

Solution steps

\int_{0}^{1} \frac{x - x}{3} d x

Take the constant out:

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

= \frac{1}{3} \cdot \int_{0}^{1} x - x 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

= \frac{1}{3} (\int_{0}^{1} x d x - \int_{0}^{1} x d x)

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

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

= \frac{1}{3} (\frac{1}{2} - \frac{2}{3})

Simplify

\frac{1}{3} (\frac{1}{2} - \frac{2}{3}) : - \frac{1}{18}

= - \frac{1}{18}

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sec (\frac{π}{3}) tan (\frac{π}{3})

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What is the integral from 0 to 1 of (x-sqrt(x))/3 ?
The integral from 0 to 1 of (x-sqrt(x))/3 is -1/18