Evaluate the integral.

${\displaystyle \underset{\phantom{}}{\overset{\phantom{}}{}}}\frac{12dx}{(x+1)(x^2+x)}$

Consider the shown work to find the partial fraction decomposition of the integrand.

$\frac{12}{(x+1)(x^2+x)}=\frac{12}{x(x+1{)}^2}=\frac{A}{x}+\frac{Bx}{(x+1{)}^2}$

$A=12,B=-12$

The partial fraction decomposition is $\frac{12}{(x+1)(x^2+x)}=\frac{12}{x}-\frac{12x}{(x+1{)}^2}.$

Identify any errors in the work shown.

$($Select all that apply.$)$

The denominator of the fraction was incorrectly factored.

No errors exist in the work shown.

The $\frac{Bx}{(x+1{)}^2}$ term should not have an $x$ in the numerator.

The values of A and $B$ were calculated incorrectly.

The partial fraction decomposition omitted a $\frac{C}{x+1}$ term.