Questions: [ int(csc x-tan x)^2 d x ]

Solution Steps

We start with the integral \[ \int(\csc x - \tan x)^{2} \, dx. \] Expanding the integrand, we have: \[ (\csc x - \tan x)^{2} = \csc^{2} x - 2 \csc x \tan x + \tan^{2} x. \]

Using the trigonometric identities \(\csc^{2} x = 1 + \cot^{2} x\) and \(\tan^{2} x = \sec^{2} x - 1\), we can rewrite the expanded integrand: \[ \csc^{2} x - 2 \csc x \tan x + \tan^{2} x = \tan^{2} x - 2 \csc x \tan x + \csc^{2} x. \]

Now we integrate the simplified expression term by term: \[ \int \left( \tan^{2} x - 2 \csc x \tan x + \csc^{2} x \right) \, dx. \] The integral results in: \[ -x + \log(\sin x - 1) - \log(\sin x + 1) + \frac{\sin x}{\cos x} - \frac{\cos x}{\sin x}. \]

Final Answer