Evaluate the integral.
\[
\int_{\pi / 4}^{3 \pi / 4} 4 \csc \theta \cot \theta d \theta
\]

Answer

Expert–verified

Hide Steps

Answer

So, the final answer is \(\boxed{0}\).

Steps

Step 1 ：We are given the integral \(\int_{\pi / 4}^{3 \pi / 4} 4 \csc \theta \cot \theta d \theta\).

Step 2 ：The integral of \(\csc(x) \cdot \cot(x)\) is \(-\csc(x)\). So, the integral of \(4 \cdot \csc(x) \cdot \cot(x)\) is \(-4 \cdot \csc(x)\).

Step 3 ：We need to evaluate this from \(\pi/4\) to \(3\pi/4\).

Step 4 ：After evaluating, we find that the integral equals to 0.

Step 5 ：So, the final answer is \(\boxed{0}\).