What is the integral from-1 to 512 of x^{-1/3} ?

The integral from-1 to 512 of x^{-1/3} is 189/2

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integral from-1 to 512 of x^{-1/3}

\int_{- 1}^{512} x^{- \frac{1}{3}} d x

\frac{189}{2}

Decimal

94.5

Solution steps

\int_{- 1}^{512} x^{- \frac{1}{3}} d x

Find undefined (singularity) points:

x = 0

If exist

b, a < b < c, f (b) =

undefined

, \int_{a}^{c} f (x) d x = \int_{a}^{b} f (x) d x + \int_{b}^{c} f (x) d x

= \int_{- 1}^{0} x^{- \frac{1}{3}} d x + \int_{0}^{512} x^{- \frac{1}{3}} d x

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

\int_{0}^{512} x^{- \frac{1}{3}} d x = 96

= - \frac{3 ( - 1 ) ^{\frac{2}{3}}}{2} + 96

Simplify

- \frac{3 ( - 1 ) ^{\frac{2}{3}}}{2} + 96 : \frac{189}{2}

= \frac{189}{2}

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What is the integral from-1 to 512 of x^{-1/3} ?
The integral from-1 to 512 of x^{-1/3} is 189/2