Choose the correct Answer of the Following Questions : `int_0^(pi/2) dx/(1 + tanx)`

Options

A. `pi`
B. `pi/3`
C. `pi/2`
D. `pi/4`

Answer

To solve the integral \( I = \int_0^{\frac{\pi}{2}} \frac{dx}{1 + \tan x} \), we can use a property of definite integrals.

### Step 1: Set up the integral
Let
\[ I = \int_0^{\frac{\pi}{2}} \frac{dx}{1 + \tan x} \]

### Step 2: Use the substitution \( x = \frac{\pi}{2} - t \)
We can use the substitution \( x = \frac{\pi}{2} - t \). Then, \( dx = -dt \) and when \( x = 0 \), \( t = \frac{\pi}{2} \) and when \( x = \frac{\pi}{2} \), \( t = 0 \). Thus, we have:
\[ I = \int_{\frac{\pi}{2}}^0 \frac{-dt}{1 + \tan\left(\frac{\pi}{2} - t\right)} \]

### Step 3: Simplify the integral
Using the identity \( \tan\left(\frac{\pi}{2} - t\right) = \cot t \), we can rewrite the integral:
\[ I = \int_{\frac{\pi}{2}}^0 \frac{-dt}{1 + \cot t} = \int_0^{\frac{\pi}{2}} \frac{dt}{1 + \cot t} \]

### Step 4: Rewrite \( \cot t \)
We know that \( \cot t = \frac{1}{\tan t} \), so:
\[ 1 + \cot t = 1 + \frac{1}{\tan t} = \frac{\tan t + 1}{\tan t} \]
Thus:
\[ \frac{1}{1 + \cot t} = \frac{\tan t}{\tan t + 1} \]

### Step 5: Substitute back into the integral
Now we can substitute this back into our integral:
\[ I = \int_0^{\frac{\pi}{2}} \frac{\tan t}{\tan t + 1} dt \]

### Step 6: Combine the two integrals
Now we have two expressions for \( I \):
1. \( I = \int_0^{\frac{\pi}{2}} \frac{dx}{1 + \tan x} \)
2. \( I = \int_0^{\frac{\pi}{2}} \frac{\tan x}{\tan x + 1} dx \)

Adding these two integrals:
\[ 2I = \int_0^{\frac{\pi}{2}} \left( \frac{1}{1 + \tan x} + \frac{\tan x}{\tan x + 1} \right) dx \]

### Step 7: Simplify the combined integral
The expression simplifies to:
\[ 2I = \int_0^{\frac{\pi}{2}} \frac{1 + \tan x}{1 + \tan x} dx = \int_0^{\frac{\pi}{2}} 1 \, dx \]
This integral evaluates to:
\[ \int_0^{\frac{\pi}{2}} 1 \, dx = \left[ x \right]_0^{\frac{\pi}{2}} = \frac{\pi}{2} \]

### Step 8: Solve for \( I \)
Thus, we have:
\[ 2I = \frac{\pi}{2} \implies I = \frac{\pi}{4} \]

### Conclusion
The value of the integral \( \int_0^{\frac{\pi}{2}} \frac{dx}{1 + \tan x} \) is:
\[ \boxed{\frac{\pi}{4}} \]