If area bounded by `y =f(x)` y - axis and the line `2y =pi (x+1)` where `f(x) = sin^(-1) x+ cos^(-1) x + tan^(-1) x + tan^(-1) ""(1)/x` is :

Options

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

Answer

To find the area bounded by the curve \( y = f(x) \), the y-axis, and the line \( 2y = \pi(x + 1) \), we will follow these steps:

### Step 1: Understand the function \( f(x) \)
The function is given as:
\[ f(x) = \sin^{-1}(x) + \cos^{-1}(x) + \tan^{-1}(x) + \tan^{-1}\left(\frac{1}{x}\right) \]

### Step 2: Simplify \( f(x) \)
Using the identity \( \sin^{-1}(x) + \cos^{-1}(x) = \frac{\pi}{2} \) for \( x \in [-1, 1] \), we can simplify \( f(x) \):
\[ f(x) = \frac{\pi}{2} + \tan^{-1}(x) + \tan^{-1}\left(\frac{1}{x}\right) \]
Using the identity \( \tan^{-1}(x) + \tan^{-1}\left(\frac{1}{x}\right) = \frac{\pi}{2} \) for \( x > 0 \):
\[ f(x) = \frac{\pi}{2} + \frac{\pi}{2} = \pi \quad \text{for } x > 0 \]

### Step 3: Determine the line equation
The line given is:
\[ 2y = \pi(x + 1) \implies y = \frac{\pi}{2}(x + 1) \]

### Step 4: Find intersections with the y-axis and x-axis
1. **Intersection with the y-axis**: Set \( x = 0 \):
\[ y = \frac{\pi}{2}(0 + 1) = \frac{\pi}{2} \quad \text{(Point A: (0, π/2))} \]

2. **Intersection with the x-axis**: Set \( y = 0 \):
\[ 0 = \frac{\pi}{2}(x + 1) \implies x + 1 = 0 \implies x = -1 \quad \text{(Point B: (-1, 0))} \]

### Step 5: Identify the area bounded by the lines and the y-axis
The area we need to find is bounded by the line \( y = \frac{\pi}{2}(x + 1) \), the y-axis, and the line \( y = \pi \).

### Step 6: Calculate the area of triangle ABC
The triangle formed by points A (0, π/2), B (-1, 0), and the intersection of the line with \( y = \pi \).

1. **Height of the triangle**: The height from point A to the line \( y = \pi \) is:
\[ \text{Height} = \pi - \frac{\pi}{2} = \frac{\pi}{2} \]

2. **Base of the triangle**: The base is the distance between points A and B along the x-axis:
\[ \text{Base} = 0 - (-1) = 1 \]

3. **Area of triangle**:
\[ \text{Area} = \frac{1}{2} \times \text{Base} \times \text{Height} = \frac{1}{2} \times 1 \times \frac{\pi}{2} = \frac{\pi}{4} \]

### Final Answer
The area bounded by \( y = f(x) \), the y-axis, and the line \( 2y = \pi(x + 1) \) is:
\[ \boxed{\frac{\pi}{4}} \]