Evaluate the integral.

${\displaystyle \underset{\phantom{}}{\overset{\phantom{}}{}}}8co{t}^2(x)cs{c}^4(x)dx$

Consider the shown work.

${\displaystyle \underset{\phantom{}}{\overset{\phantom{}}{}}}8co{t}^2(x)cs{c}^4(x)dx={\displaystyle \underset{\phantom{}}{\overset{\phantom{}}{}}}8co{t}^2(x)(1-co{t}^2(x))cs{c}^2(x)dx={\displaystyle \underset{\phantom{}}{\overset{\phantom{}}{}}}8(co{t}^2(x)-co{t}^4(x))cs{c}^2(x)dx$

Let $u=cot(x)$ and $du=-cs{c}^2(x)dx.$

${\displaystyle \underset{\phantom{}}{\overset{\phantom{}}{}}}8(co{t}^2(x)-co{t}^4(x))cs{c}^2(x)dx=-8{\displaystyle \underset{\phantom{}}{\overset{\phantom{}}{}}}(u^2-u^4)du$

$=-\frac{8}{3}{u}^3+\frac{8}{5}{u}^5+C$

$=-\frac{8}{3}co{t}^3(x)+\frac{8}{5}co{t}^5(x)+C$

Identify the error in the work shown.

The substitution $cs{c}^2(x)=1-co{t}^2(x)$ is not a correct trigonometric identity.

When applying the substitution method, an inappropriate expression was used for $u.$

The power rule for integrals was not applied correctly.

The resulting integral after applying the substitution method is incorrect.

No errors exist in the work shown.

Evaluate the integral.

${\displaystyle \underset{\phantom{}}{\overset{\phantom{}}{}}}8co{t}^2(x)cs{c}^4(x)dx$

$($Express numbers in exact form. Use symbolic notation and fractions as needed. Use $C$ to represent the arbitrary constant.$)$