What is the integral from 1 to 10 of 1/((x-2)^{1/3)} ?

The integral from 1 to 10 of 1/((x-2)^{1/3)} is 9/2

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integral from 1 to 10 of 1/((x-2)^{1/3)}

\int_{1}^{10} \frac{1}{( x - 2 ) ^{\frac{1}{3}}} d x

\frac{9}{2}

Solution steps

\int_{1}^{10} \frac{1}{( x - 2 ) ^{\frac{1}{3}}} d x

Find undefined (singularity) points:

x = 2

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}^{2} \frac{1}{( x - 2 ) ^{\frac{1}{3}}} d x + \int_{2}^{10} \frac{1}{( x - 2 ) ^{\frac{1}{3}}} d x

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

\int_{2}^{10} \frac{1}{( x - 2 ) ^{\frac{1}{3}}} d x = 6

= - \frac{3 ( - 1 ) ^{\frac{2}{3}}}{2} + 6

Simplify

- \frac{3 ( - 1 ) ^{\frac{2}{3}}}{2} + 6 : \frac{9}{2}

= \frac{9}{2}

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What is the integral from 1 to 10 of 1/((x-2)^{1/3)} ?
The integral from 1 to 10 of 1/((x-2)^{1/3)} is 9/2