Answer
$$\int^{1}_0\frac{dr}{\sqrt[3]{(7-5r)^2}}=-\frac{3}{5}(\sqrt[3]2-\sqrt[3]7)$$
Work Step by Step
$$A=\int^{1}_0\frac{dr}{\sqrt[3]{(7-5r)^2}}=\int^{1}_0\frac{dr}{(7-5r)^{2/3}}$$
We set $a=7-5r$, which means $$da=-5dr$$ $$dr=-\frac{1}{5}da$$
For $r=0$, $a=7$ and for $r=1$, $a=2$
Therefore,
$$A=-\frac{1}{5}\int^{2}_7\frac{da}{a^{2/3}}=-\frac{1}{5}\int^{2}_7a^{-2/3}da$$ $$A=\Big(-\frac{1}{5}\times\frac{a^{1/3}}{\frac{1}{3}}\Big)\Big]^2_7$$ $$A=\Big(-\frac{3}{5}\sqrt[3]a\Big)\Big]^2_7$$ $$A=-\frac{3}{5}(\sqrt[3]2-\sqrt[3]7)$$
