`int_(0)^(pi//4) cot x. " cosec"^(2) x dx`

Answer

To solve the integral \( I = \int_{0}^{\frac{\pi}{4}} \cot x \csc^2 x \, dx \), we can follow these steps:

### Step 1: Rewrite the Integral
We can express \( \csc^2 x \) in terms of \( \cot x \):
\[ I = \int_{0}^{\frac{\pi}{4}} \cot x \csc^2 x \, dx \]
We know that \( \csc^2 x = 1 + \cot^2 x \), but for our purposes, we can keep it as is.

### Step 2: Use Substitution
Let \( t = \csc x \). Then, we need to find \( dt \):
\[ \frac{dt}{dx} = -\csc x \cot x \implies dt = -\csc x \cot x \, dx \implies dx = -\frac{dt}{\csc x \cot x} \]
Thus, we can rewrite the integral:
\[ I = \int \cot x \cdot \csc^2 x \cdot \left(-\frac{dt}{\csc x \cot x}\right) \]
This simplifies to:
\[ I = -\int \csc x \, dt \]

### Step 3: Change the Limits of Integration
Now, we need to change the limits of integration according to our substitution:
- When \( x = 0 \), \( t = \csc(0) = \infty \)
- When \( x = \frac{\pi}{4} \), \( t = \csc\left(\frac{\pi}{4}\right) = \sqrt{2} \)

Thus, the integral becomes:
\[ I = -\int_{\infty}^{\sqrt{2}} \csc x \, dt \]

### Step 4: Evaluate the Integral
Now we can evaluate the integral:
\[ I = -\left[ -\frac{1}{2} t^2 \right]_{\infty}^{\sqrt{2}} = \frac{1}{2} \left( \left(\sqrt{2}\right)^2 - \infty^2 \right) \]
This simplifies to:
\[ I = \frac{1}{2} \left( 2 - \infty \right) = \frac{1}{2} \left( -\infty \right) = -\infty \]

### Final Result
Thus, the value of the integral is:
\[ I = -\infty \]