Consider the integral ${\displaystyle \underset{\phantom{}}{\overset{\phantom{}}{}}}\frac{1}{t^2(t-1)}dt.$

$($a$)(8$ points$)$ It can be shown that there are constants $A,B,C$ such the rational function in the integrand decomposes into partial fractions as

$\frac{1}{t^2(t-1)}=\frac{A}{t}\frac{B}{t^2}\frac{C}{t-1}$

Find the values of the constants $A,B,C.($Show all of your work.$)$

$($b$)(6$ points$)$ Evaluate ${\displaystyle \underset{\phantom{}}{\overset{\phantom{}}{}}}\frac{1}{t^2(t-1)}dt.$