Integral Exercise 28

We are asked to evaluate the following integral:

$$ \int x^{-\frac{1}{3}} \ dx $$

This is a simple power function, so it can be handled using the standard power rule for integration:

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

Applying the rule with \( n = -\frac{1}{3} \), we get:

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

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

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

So, the integral evaluates to:

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

And that completes the solution.