The value of `int_(pi//4)^(pi//2) cot theta cosec^(2) theta d theta` is

Options

A. `- (1)/(2)`
B. `1/2`
C. `1/3`
D. `-(1)/(3)`

Answer

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

### Step 1: Rewrite the integral
We start with the integral:
\[ \int_{\frac{\pi}{4}}^{\frac{\pi}{2}} \cot \theta \csc^2 \theta \, d\theta \]

### Step 2: Use substitution
Let \( u = \cot \theta \). Then, we need to find \( du \):
\[ \frac{du}{d\theta} = -\csc^2 \theta \quad \Rightarrow \quad du = -\csc^2 \theta \, d\theta \quad \Rightarrow \quad d\theta = -\frac{du}{\csc^2 \theta} \]

### Step 3: Change the limits of integration
When \( \theta = \frac{\pi}{4} \):
\[ u = \cot\left(\frac{\pi}{4}\right) = 1 \]
When \( \theta = \frac{\pi}{2} \):
\[ u = \cot\left(\frac{\pi}{2}\right) = 0 \]
Thus, the limits change from \( \theta: \frac{\pi}{4} \to \frac{\pi}{2} \) to \( u: 1 \to 0 \).

### Step 4: Substitute into the integral
Substituting \( u \) and changing the limits, we have:
\[ \int_{1}^{0} u (-du) = \int_{0}^{1} u \, du \]

### Step 5: Evaluate the integral
Now we can evaluate the integral:
\[ \int_{0}^{1} u \, du = \left[ \frac{u^2}{2} \right]_{0}^{1} = \frac{1^2}{2} - \frac{0^2}{2} = \frac{1}{2} \]

### Final Answer
Thus, the value of the integral \( \int_{\frac{\pi}{4}}^{\frac{\pi}{2}} \cot \theta \csc^2 \theta \, d\theta \) is:
\[ \frac{1}{2} \]

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