Say I have to calculate $$\int_{-1}^1 \frac{\text{d}x}{\sqrt[3]{x}}$$
I know I have to split it becasue $0$ belongs to the integration path, hence I have to check for the well-definition of the function at zero. What is the correct way to split the integral? I thought of two ways, but maybe they are both wrong.
- $(1)$
$$\int_{-1}^1 = \lim_{\epsilon \to 0} \left(\int_{-1}^{\epsilon} + \int_{\epsilon}^1\right)$$
- $(2)$
$$\int_{-1}^1 = \lim_{\epsilon \to 0} \left( \int_{-1}^{0 - \epsilon} + \int_{0 - \epsilon}^{0+\epsilon} + \int_{0 + \epsilon}^1\right)$$
Thank you!