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

The integral of (x-1)^{-2/3} is 3(x-1)^{1/3}+C

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

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

3 (x - 1)^{\frac{1}{3}} + C

Solution steps

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

Apply exponent rule:

a^{- b} = \frac{1}{a ^{b}}

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

Apply u-substitution

: \int \frac{1}{u ^{\frac{2}{3}}} d u

= \int \frac{1}{u ^{\frac{2}{3}}} d u

Apply the Power Rule

: 3 u^{\frac{1}{3}}

= 3 u^{\frac{1}{3}}

Substitute back

u = x - 1

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

Add a constant to the solution

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

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What is the integral of (x-1)^{-2/3} ?
The integral of (x-1)^{-2/3} is 3(x-1)^{1/3}+C