Evaluate this integral $$\int_0^1(x^2-x^3)^{-\frac 1 3}dx$$ by using a barbell-shaped contour with shrinking end surrounding 0 and 1, together with a large circle containing the barbell.    The branch cut connects 0 and 1. Let the value of $f(z)=z^{-\frac 2 3}(1-z)^{-\frac1 3}$ over the branch cut be positive. When a point over the branch cut moves along the red curve to the corresponding point under the branch cut, $\Delta \arg(z^{-\frac 2 3})=e^{-i{\frac {3\pi}{4}}}$ and $\Delta \arg\big((1-z)^{-\frac1 3}\big)=0$. So $e^{-i\frac {4\pi} 3}$ should be multiplied under the branch cut.   The integral over the outer circle equals:$$-2\pi iRes(f(z),\infty)=2\pi iRes(f(\frac 1 t)  \cdot \frac 1 {t^2},0)=2\pi iRes(\frac 1 t (t-1)^{-\frac 1 3})=2\pi i(-1)^{-\frac 1 3}.$$ Because of the branch we chose, $(-1)^{-\frac 1 3}=e^{-i{\frac \pi 3}}.$ We get $$(1-e^{-i{\frac {4\pi} 3}})\int_0^1(x^2-x^3)^{-\frac 1 3}dx + 2\pi ie^{-i{\frac \pi 3}}=0.$$ So,$$\int_0^1(x^2-x^3)^{-\frac 1 3}dx=\frac {2\pi} {\sqrt 3}.$$
How do I go about evaluating the following by contour integration?  $$ \int^1_0 \frac{dx}{(x^{2} - x^{3})^{1/3}} $$  The question does not fit in the standard form of : $\int^{2\pi}_0$ or $\int^\infty_{-\infty}$  etc.  I tried substituting $ x = cos(\theta) $ but got stuck.
"Calling dibs" - what does it imply? The term "to have dibs on something" or "to call dibs on something" plays a recurring role in American film and television (e.g. ), so it gets exported a lot.   describes "dibs" as     [...] a common, "informal" convention to reserve or declare full or partial ownership of a community resource, such as a chair [...]   From the usages in the media I have seen, this convention also seems to contain some moral concept which is not described in dictionaries. For me, it looks like by calling dibs (or ), the caller actually receives the (implicit, moral) right to that resource, and this typically goes undisputed by their rivals.  Can you confirm the existence of this concept? Is this something that varies between groups of people (or maybe between regions), or is there a concensus on how to respond to this? 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