Integral of x^(-2/3)

We’re asked to evaluate the integral of \( x^{-2/3} \):

$$ \int x^{- \frac{2}{3}} \ dx $$

This is a standard power function, so we apply the power rule for integration:

$$ \int x^n \ dx = \frac{x^{n+1}}{n+1} + c, \quad \text{for } n \ne -1 $$

Here, \( n = -\frac{2}{3} \), so we proceed as follows:

$$ \int x^{- \frac{2}{3}} \ dx = \frac{x^{- \frac{2}{3} + 1}}{- \frac{2}{3} + 1} + c $$

Now simplify the exponent and the denominator:

$$ = \frac{x^{\frac{-2 + 3}{3}}}{\frac{-2 + 3}{3}} + c $$

$$ = \frac{x^{\frac{1}{3}}}{\frac{1}{3}} + c $$

$$ = 3x^{\frac{1}{3}} + c $$

Therefore, the solution to the integral is:

$$ \int x^{- \frac{2}{3}} \ dx = 3x^{\frac{1}{3}} + c $$