`int_(-pi//4)^(pi//4) "cosec"^(2) x dx`

Answer

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

### Step 1: Use the property of definite integrals
We can use the property of definite integrals that states:
\[ \int_{-a}^{a} f(x) \, dx = 2 \int_{0}^{a} f(x) \, dx \]
In our case, \( f(x) = \csc^2 x \) and \( a = \frac{\pi}{4} \). Therefore, we can rewrite the integral as:
\[ \int_{-\frac{\pi}{4}}^{\frac{\pi}{4}} \csc^2 x \, dx = 2 \int_{0}^{\frac{\pi}{4}} \csc^2 x \, dx \]

### Step 2: Integrate \( \csc^2 x \)
The integral of \( \csc^2 x \) is known:
\[ \int \csc^2 x \, dx = -\cot x + C \]
Thus, we can evaluate the integral from \( 0 \) to \( \frac{\pi}{4} \):
\[ \int_{0}^{\frac{\pi}{4}} \csc^2 x \, dx = \left[-\cot x\right]_{0}^{\frac{\pi}{4}} \]

### Step 3: Evaluate the limits
Now we need to evaluate \( -\cot x \) at the limits \( 0 \) and \( \frac{\pi}{4} \):
1. At \( x = \frac{\pi}{4} \):
\[ -\cot\left(\frac{\pi}{4}\right) = -1 \]
2. At \( x = 0 \):
\[ -\cot(0) = -\infty \quad (\text{cotangent is undefined at } 0) \]

### Step 4: Combine the results
Putting it all together, we have:
\[ \int_{0}^{\frac{\pi}{4}} \csc^2 x \, dx = -1 - (-\infty) = \infty \]
Thus,
\[ 2 \int_{0}^{\frac{\pi}{4}} \csc^2 x \, dx = 2 \cdot \infty = \infty \]

### Conclusion
Therefore, the value of the integral \( \int_{-\frac{\pi}{4}}^{\frac{\pi}{4}} \csc^2 x \, dx \) is:
\[ \text{The integral diverges to } \infty. \]