\( \int _{}^{}(\csc (2x)+\cot (2x))^{2} d \)
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To integrate \((\csc(2x) + \cot(2x))^2\), we can first simplify the expression. We know that \(\csc(2x) = \frac{1}{\sin(2x)}\) and \(\cot(2x) = \frac{\cos(2x)}{\sin(2x)}\). Thus, the expression can be rewritten as: \[ (\csc(2x) + \cot(2x))^2 = \left(\frac{1 + \cos(2x)}{\sin(2x)}\right)^2 = \frac{(1 + \cos(2x))^2}{\sin^2(2x)} \] Using the identity \(1 + \cos(2x) = 2\cos^2(x)\), we get: \[ (1 + \cos(2x))^2 = (2\cos^2(x))^2 = 4\cos^4(x) \] This gives us: \[ \frac{(1 + \cos(2x))^2}{\sin^2(2x)} = \frac{4\cos^4(x)}{\sin^2(2x)} \] Therefore, we can rewrite the integral as: \[ \int \frac{4\cos^4(x)}{\sin^2(2x)} \, dx \] Noticing that \(\sin(2x) = 2\sin(x)\cos(x)\), we have \(\sin^2(2x) = 4\sin^2(x)\cos^2(x)\). This allows us to express the integral as: \[ \int \frac{4\cos^4(x)}{4\sin^2(x)\cos^2(x)} \, dx = \int \frac{\cos^2(x)}{\sin^2(x)} \, dx = \int \cot^2(x) \, dx \] The integral of \( \cot^2(x) \) can be computed using the identity \(\cot^2(x) = \csc^2(x) - 1\): \[ \int \cot^2(x) \, dx = \int (\csc^2(x) - 1) \, dx = -\cot(x) - x + C \] Thus, we find: \[ \int (\csc(2x) + \cot(2x))^2 \, dx = -\frac{1}{2}\cot(2x) - \frac{1}{2}x + C \] And that’s how you handle the integral! Happy integrating!
