Answer
$$3$$
Work Step by Step
Simplify the improper integral $\int_{0}^{1} \dfrac{1}{(x-1)^{2/3}} dx$
Let us consider that $I=\int_{0}^{1} \dfrac{1}{(x-1)^{2/3}} dx$
Plug $a=x-1 \implies da=dx$
Integrate the integral by using limits.
$I=\int_0^1 \dfrac{1}{a^{2/3}} da \\=|3a^{1/3}|_0^1\\=|3 \sqrt[3]{x-1}|_0^1 \\=0-(-3) \\=3$
