What is the integral from 0 to 1 of sqrt(x)(x^2-1) ?

The integral from 0 to 1 of sqrt(x)(x^2-1) is -8/21

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integral from 0 to 1 of sqrt(x)(x^2-1)

\int_{0}^{1} x (x^{2} - 1) d x

- \frac{8}{21}

Decimal

- 0.38095 \dots

Solution steps

\int_{0}^{1} x (x^{2} - 1) d x

Expand

x (x^{2} - 1) : x^{\frac{5}{2}} - x

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

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

\int_{0}^{1} x^{\frac{5}{2}} d x = \frac{2}{7}

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

= \frac{2}{7} - \frac{2}{3}

Simplify

\frac{2}{7} - \frac{2}{3} : - \frac{8}{21}

= - \frac{8}{21}

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What is the integral from 0 to 1 of sqrt(x)(x^2-1) ?
The integral from 0 to 1 of sqrt(x)(x^2-1) is -8/21