`int_(pi//4)^(pi//2)cot^(2)x dx =`

Options

A. `(pi-4)/(4)`
B. `1-(pi)/(4)`
C. `1-(pi)/(2)`
D. `(pi)/(8)`

Answer

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

### Step 1: Rewrite the integrand
We know that:
\[ \cot^2 x = \csc^2 x - 1 \]
Thus, we can rewrite the integral as:
\[ \int_{\frac{\pi}{4}}^{\frac{\pi}{2}} \cot^2 x \, dx = \int_{\frac{\pi}{4}}^{\frac{\pi}{2}} (\csc^2 x - 1) \, dx \]

### Step 2: Split the integral
Now, we can split the integral into two parts:
\[ \int_{\frac{\pi}{4}}^{\frac{\pi}{2}} \cot^2 x \, dx = \int_{\frac{\pi}{4}}^{\frac{\pi}{2}} \csc^2 x \, dx - \int_{\frac{\pi}{4}}^{\frac{\pi}{2}} 1 \, dx \]

### Step 3: Integrate each part
1. The integral of \( \csc^2 x \) is:
\[ \int \csc^2 x \, dx = -\cot x \]
Therefore,
\[ \int_{\frac{\pi}{4}}^{\frac{\pi}{2}} \csc^2 x \, dx = \left[-\cot x\right]_{\frac{\pi}{4}}^{\frac{\pi}{2}} = -\cot\left(\frac{\pi}{2}\right) + \cot\left(\frac{\pi}{4}\right) \]

Since \( \cot\left(\frac{\pi}{2}\right) = 0 \) and \( \cot\left(\frac{\pi}{4}\right) = 1 \), this becomes:
\[ -0 + 1 = 1 \]

2. The integral of \( 1 \) is:
\[ \int 1 \, dx = x \]
Thus,
\[ \int_{\frac{\pi}{4}}^{\frac{\pi}{2}} 1 \, dx = \left[x\right]_{\frac{\pi}{4}}^{\frac{\pi}{2}} = \frac{\pi}{2} - \frac{\pi}{4} = \frac{\pi}{4} \]

### Step 4: Combine the results
Now we combine the results from the two integrals:
\[ \int_{\frac{\pi}{4}}^{\frac{\pi}{2}} \cot^2 x \, dx = 1 - \frac{\pi}{4} \]

### Final Answer
Thus, the final answer is:
\[ \int_{\frac{\pi}{4}}^{\frac{\pi}{2}} \cot^2 x \, dx = 1 - \frac{\pi}{4} \]

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