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

The integral from 0 to 2 of 1/((x-1)^{2/3)} is 6

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

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

6

Solution steps

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

Find undefined (singularity) points:

x = 1

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

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

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

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

Simplify

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

= 6

x \to 0 - lim ((sin (x))^{x})

(ln (2))^{^{'}}

\frac{d}{d x} (x + 1 (x + 1))

y = x^{2} - 21, y = 7 - 3 x

\frac{10 x}{x ^{2} - 6}

What is the integral from 0 to 2 of 1/((x-1)^{2/3)} ?
The integral from 0 to 2 of 1/((x-1)^{2/3)} is 6