Find the general antiderivative ʃ x^(-2/3) dx. Then verify by differentiation  Find the general antiderivative ʃ x^(-2/3) dx. Then verify by differentiation.  

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embizze | High School Teacher | (Level 1) Educator Emeritus

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Find `int x^(-2/3)dx` :

We use the general power rule: `int u^ndu=(u^(n+1))/(n+1)+C,n != 1`

Here `u=x,du=dx,n=-2/3` So

`int x^(-2/3)dx=(x^(-2/3+1))/(-2/3+1)+C=(x^(1/3))/(1/3)+C=3x^(1/3)+C`

Differentiating we get:

`d/(dx)[3x^(1/3)+C]=(3)1/3x^(-2/3)=x^(-2/3)` as required.

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`int x^(-2/3)dx=3x^(1/3)+C`

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