k) \( \int \frac{d x}{\sin ^{2} \frac{x}{4}} \)

To solve the integral \( \int \frac{d x}{\sin ^{2} \frac{x}{4}} \), we can use the identity that relates \( \sin^2 \) to cosecant: \( \frac{1}{\sin^2 \theta} = \csc^2 \theta \). This transforms our integral into: \[ \int \csc^2 \frac{x}{4} \, dx \] The integral of \( \csc^2 u \) is known to be \(-\cot u + C\), where \( C \) is the integration constant. We need to adjust for our variable by making a substitution. Let \( u = \frac{x}{4} \), then \( dx = 4 du \). Now we can rewrite our integral: \[ 4 \int \csc^2 u \, du = 4 \left(-\cot u + C\right) = -4 \cot \frac{x}{4} + C \] Thus, the solution to the integral is: \[ \int \frac{d x}{\sin ^{2} \frac{x}{4}} = -4 \cot \frac{x}{4} + C \] So, \( -4 \cot \frac{x}{4} + C \) is your answer!