What is the integral from 0 to 1 of x^{1/3}-x^2 ?

The integral from 0 to 1 of x^{1/3}-x^2 is 5/12

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integral from 0 to 1 of x^{1/3}-x^2

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

\frac{5}{12}

Decimal

0.41666 \dots

Solution steps

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

\int_{0}^{1} x^{\frac{1}{3}} d x = \frac{3}{4}

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

= \frac{3}{4} - \frac{1}{3}

Simplify

\frac{3}{4} - \frac{1}{3} : \frac{5}{12}

= \frac{5}{12}

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What is the integral from 0 to 1 of x^{1/3}-x^2 ?
The integral from 0 to 1 of x^{1/3}-x^2 is 5/12