Select all expressions matching the value of indefinite integral int (-dt)/(sin^2)(t)/(4) -cot^2(t)/(4)+C cot(t)/(16)+C 4cot(t)/(4)+C

To solve the indefinite integral \(\int \frac{-dt}{\sin^2 \frac{t}{4}}\), we can use a trigonometric identity and substitution.<br /><br />First, recall the identity:<br />\[<br />\csc^2 x = \frac{1}{\sin^2 x}<br />\]<br /><br />So, we can rewrite the integral as:<br />\[<br />\int \frac{-dt}{\sin^2 \frac{t}{4}} = -\int \csc^2 \frac{t}{4} \, dt<br />\]<br /><br />Next, use the substitution \( u = \frac{t}{4} \), which implies \( du = \frac{1}{4} dt \) or \( dt = 4 du \).<br /><br />Substitute \( u \) and \( dt \) into the integral:<br />\[<br />-\int \csc^2 \frac{t}{4} \, dt = -\int \csc^2 u \cdot 4 \, du = -4 \int \csc^2 u \, du<br />\]<br /><br />The integral of \(\csc^2 u\) is \(-\cot u\):<br />\[<br />-4 \int \csc^2 u \, du = -4 (-\cot u) + C = 4 \cot u + C<br />\]<br /><br />Now, substitute back \( u = \frac{t}{4} \):<br />\[<br />4 \cot \left( \frac{t}{4} \right) + C<br />\]<br /><br />Thus, the correct expression matching the value of the indefinite integral is:<br />\[<br />4 \cot \frac{t}{4} + C<br />\]<br /><br />So, the correct answer is:<br />\[<br />4 \cot \frac{t}{4} + C<br />\]