What is the integral from 1 to 3 of 1/((x-2)^{5/3)} ?

The integral from 1 to 3 of 1/((x-2)^{5/3)} is diverges

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integral from 1 to 3 of 1/((x-2)^{5/3)}

\int_{1}^{3} \frac{1}{( x - 2 ) ^{\frac{5}{3}}} d x

diverges

Solution steps

\int_{1}^{3} \frac{1}{( x - 2 ) ^{\frac{5}{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{5}{3}}} d x + \int_{2}^{3} \frac{1}{( x - 2 ) ^{\frac{5}{3}}} d x

\int_{1}^{2} \frac{1}{( x - 2 ) ^{\frac{5}{3}}} d x =

diverges

If part of the expression diverges, then entire expression diverges

= diverges

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\int_{0}^{1} (7 - \frac{21}{2} x)^{2} d x

What is the integral from 1 to 3 of 1/((x-2)^{5/3)} ?
The integral from 1 to 3 of 1/((x-2)^{5/3)} is diverges