Answer
$$ - 4\cot x + C$$
Work Step by Step
$$\eqalign{
& \int {4{{\csc }^2}x} dx \cr
& {\text{use multiple constant rule }} \cr
& \int {4{{\csc }^2}x} dx = 4\int {{{\csc }^2}x} dx \cr
& {\text{integrate by using the Basic Trigonometric integral }}\int {{{\csc }^2}x} dx = - \cot x + C \cr
& then \cr
& \int {4{{\csc }^2}x} dx = - 4\cot x + C \cr} $$